Symbolic Expressions

RELATIONAL EXPRESSIONS:

We create a relational expression:

sage: x = var('x')
sage: eqn = (x-1)^2 <= x^2 - 2*x + 3
sage: eqn.subs(x == 5)
16 <= 18

Notice that squaring the relation squares both sides.

sage: eqn^2
(x - 1)^4 <= (x^2 - 2*x + 3)^2
sage: eqn.expand()
x^2 - 2*x + 1 <= x^2 - 2*x + 3

The can transform a true relational into a false one:

sage: eqn = SR(-5) < SR(-3); eqn
-5 < -3
sage: bool(eqn)
True
sage: eqn^2
25 < 9
sage: bool(eqn^2)
False

We can do arithmetic with relationals:

sage: e = x+1 <= x-2
sage: e + 2
x + 3 <= x
sage: e - 1
x <= x - 3
sage: e*(-1)
-x - 1 <= -x + 2
sage: (-2)*e
-2*x - 2 <= -2*x + 4
sage: e*5
5*x + 5 <= 5*x - 10
sage: e/5
1/5*x + 1/5 <= 1/5*x - 2/5
sage: 5/e
5/(x + 1) <= 5/(x - 2)
sage: e/(-2)
-1/2*x - 1/2 <= -1/2*x + 1
sage: -2/e
-2/(x + 1) <= -2/(x - 2)

We can even add together two relations, so long as the operators are the same:

sage: (x^3 + x <= x - 17)  + (-x <= x - 10)
x^3 <= 2*x - 27

Here they are not:

sage: (x^3 + x <= x - 17)  + (-x >= x - 10)
Traceback (most recent call last):
...
TypeError: incompatible relations

ARBITRARY SAGE ELEMENTS:

You can work symbolically with any Sage data type. This can lead to nonsense if the data type is strange, e.g., an element of a finite field (at present).

We mix Singular variables with symbolic variables:

sage: R.<u,v> = QQ[]
sage: var('a,b,c')
(a, b, c)
sage: expand((u + v + a + b + c)^2)
a^2 + 2*a*b + b^2 + 2*a*c + 2*b*c + c^2 + 2*a*u + 2*b*u + 2*c*u + u^2 + 2*a*v + 2*b*v + 2*c*v + 2*u*v + v^2

TESTS:

Test Jacobian on Pynac expressions. (trac ticket #5546)

sage: var('x,y')
(x, y)
sage: f = x + y
sage: jacobian(f, [x,y])
[1 1]

Test if matrices work (trac ticket #5546)

sage: var('x,y,z')
(x, y, z)
sage: M = matrix(2,2,[x,y,z,x])
sage: v = vector([x,y])
sage: M * v
(x^2 + y^2, x*y + x*z)
sage: v*M
(x^2 + y*z, 2*x*y)

Test if comparison bugs from trac ticket #6256 are fixed:

sage: t = exp(sqrt(x)); u = 1/t
sage: t*u
1
sage: t + u
e^(-sqrt(x)) + e^sqrt(x)
sage: t
e^sqrt(x)

Test if trac ticket #9947 is fixed:

sage: real_part(1+2*(sqrt(2)+1)*(sqrt(2)-1))
3
sage: a=(sqrt(4*(sqrt(3) - 5)*(sqrt(3) + 5) + 48) + 4*sqrt(3))/ (sqrt(3) + 5)
sage: a.real_part()
4*sqrt(3)/(sqrt(3) + 5)
sage: a.imag_part()
sqrt(abs(4*(sqrt(3) + 5)*(sqrt(3) - 5) + 48))/(sqrt(3) + 5)

class sage.symbolic.expression.Expression

Bases: sage.structure.element.CommutativeRingElement

Nearly all expressions are created by calling new_Expression_from_*, but we need to make sure this at least does not leave self._gobj uninitialized and segfault.

TESTS:

sage: sage.symbolic.expression.Expression(SR)
0
sage: sage.symbolic.expression.Expression(SR, 5)
5

We test subclassing Expression:

sage: from sage.symbolic.expression import Expression
sage: class exp_sub(Expression): pass
sage: f = function('f')
sage: t = f(x)
sage: u = exp_sub(SR, t)
sage: u.operator()
f

N(prec=None, digits=None, algorithm=None)

Return a numerical approximation this symbolic expression as either a real or complex number with at least the requested number of bits or digits of precision.

EXAMPLES:

sage: sin(x).subs(x=5).n()
-0.958924274663138
sage: sin(x).subs(x=5).n(100)
-0.95892427466313846889315440616
sage: sin(x).subs(x=5).n(digits=50)
-0.95892427466313846889315440615599397335246154396460
sage: zeta(x).subs(x=2).numerical_approx(digits=50)
1.6449340668482264364724151666460251892189499012068

sage: cos(3).numerical_approx(200)
-0.98999249660044545727157279473126130239367909661558832881409
sage: numerical_approx(cos(3),200)
-0.98999249660044545727157279473126130239367909661558832881409
sage: numerical_approx(cos(3), digits=10)
-0.9899924966
sage: (i + 1).numerical_approx(32)
1.00000000 + 1.00000000*I
sage: (pi + e + sqrt(2)).numerical_approx(100)
7.2740880444219335226246195788

TESTS:

We test the evaluation of different infinities available in Pynac:

sage: t = x - oo; t
-Infinity
sage: t.n()
-infinity
sage: t = x + oo; t
+Infinity
sage: t.n()
+infinity
sage: t = x - unsigned_infinity; t
Infinity
sage: t.n()
Traceback (most recent call last):
...
ValueError: can only convert signed infinity to RR

Some expressions cannot be evaluated numerically:

sage: n(sin(x))
Traceback (most recent call last):
...
TypeError: cannot evaluate symbolic expression numerically
sage: a = var('a')
sage: (x^2 + 2*x + 2).subs(x=a).n()
Traceback (most recent call last):
...
TypeError: cannot evaluate symbolic expression numerically

Make sure we’ve rounded up log(10,2) enough to guarantee sufficient precision (trac ticket #10164):

sage: ks = 4*10**5, 10**6
sage: all(len(str(e.n(digits=k)))-1 >= k for k in ks)
True

Order(hold=False)

Return the order of the expression, as in big oh notation.

OUTPUT:

A symbolic expression.

EXAMPLES:

sage: n = var('n')
sage: t = (17*n^3).Order(); t
Order(n^3)
sage: t.derivative(n)
Order(n^2)

To prevent automatic evaluation use the hold argument:

sage: (17*n^3).Order(hold=True)
Order(17*n^3)

abs(hold=False)

Return the absolute value of this expression.

EXAMPLES:

sage: var('x, y')
(x, y)
sage: (x+y).abs()
abs(x + y)

Using the hold parameter it is possible to prevent automatic evaluation:

sage: SR(-5).abs(hold=True)
abs(-5)

To then evaluate again, we currently must use Maxima via simplify():

sage: a = SR(-5).abs(hold=True); a.simplify()
5

TESTS:

From trac ticket #7557:

sage: var('y', domain='real')
y
sage: abs(exp(1.1*y*I)).simplify()
1
sage: var('y', domain='complex') # reset the domain for other tests
y

add(hold=False, *args)

Return the sum of the current expression and the given arguments.

To prevent automatic evaluation use the hold argument.

EXAMPLES:

sage: x.add(x)
2*x
sage: x.add(x, hold=True)
x + x
sage: x.add(x, (2+x), hold=True)
(x + 2) + x + x
sage: x.add(x, (2+x), x, hold=True)
(x + 2) + x + x + x
sage: x.add(x, (2+x), x, 2*x, hold=True)
(x + 2) + 2*x + x + x + x

To then evaluate again, we currently must use Maxima via simplify():

sage: a = x.add(x, hold=True); a.simplify()
2*x

add_to_both_sides(x)

Return a relation obtained by adding x to both sides of this relation.

EXAMPLES:

sage: var('x y z')
(x, y, z)
sage: eqn = x^2 + y^2 + z^2 <= 1
sage: eqn.add_to_both_sides(-z^2)
x^2 + y^2 <= -z^2 + 1
sage: eqn.add_to_both_sides(I)
x^2 + y^2 + z^2 + I <= (I + 1)

arccos(hold=False)

Return the arc cosine of self.

EXAMPLES:

sage: x.arccos()
arccos(x)
sage: SR(1).arccos()
0
sage: SR(1/2).arccos()
1/3*pi
sage: SR(0.4).arccos()
1.15927948072741
sage: plot(lambda x: SR(x).arccos(), -1,1)
Graphics object consisting of 1 graphics primitive

To prevent automatic evaluation use the hold argument:

sage: SR(1).arccos(hold=True)
arccos(1)

This also works using functional notation:

sage: arccos(1,hold=True)
arccos(1)
sage: arccos(1)
0

To then evaluate again, we currently must use Maxima via simplify():

sage: a = SR(1).arccos(hold=True); a.simplify()
0

TESTS:

sage: SR(oo).arccos()
Traceback (most recent call last):
...
RuntimeError: arccos_eval(): arccos(infinity) encountered
sage: SR(-oo).arccos()
Traceback (most recent call last):
...
RuntimeError: arccos_eval(): arccos(infinity) encountered
sage: SR(unsigned_infinity).arccos()
Infinity

arccosh(hold=False)

Return the inverse hyperbolic cosine of self.

EXAMPLES:

sage: x.arccosh()
arccosh(x)
sage: SR(0).arccosh()
1/2*I*pi
sage: SR(1/2).arccosh()
arccosh(1/2)
sage: SR(CDF(1/2)).arccosh()  # rel tol 1e-15
1.0471975511965976*I
sage: maxima('acosh(0.5)')
1.04719755119659...*%i

To prevent automatic evaluation use the hold argument:

sage: SR(-1).arccosh()
I*pi
sage: SR(-1).arccosh(hold=True)
arccosh(-1)

This also works using functional notation:

sage: arccosh(-1,hold=True)
arccosh(-1)
sage: arccosh(-1)
I*pi

To then evaluate again, we currently must use Maxima via simplify():

sage: a = SR(-1).arccosh(hold=True); a.simplify()
I*pi

TESTS:

sage: SR(oo).arccosh()
+Infinity
sage: SR(-oo).arccosh()
+Infinity
sage: SR(unsigned_infinity).arccosh()
+Infinity

arcsin(hold=False)

Return the arcsin of x, i.e., the number y between -pi and pi such that sin(y) == x.

EXAMPLES:

sage: x.arcsin()
arcsin(x)
sage: SR(0.5).arcsin()
0.523598775598299
sage: SR(0.999).arcsin()
1.52607123962616
sage: SR(1/3).arcsin()
arcsin(1/3)
sage: SR(-1/3).arcsin()
-arcsin(1/3)

To prevent automatic evaluation use the hold argument:

sage: SR(0).arcsin()
0
sage: SR(0).arcsin(hold=True)
arcsin(0)

This also works using functional notation:

sage: arcsin(0,hold=True)
arcsin(0)
sage: arcsin(0)
0

To then evaluate again, we currently must use Maxima via simplify():

sage: a = SR(0).arcsin(hold=True); a.simplify()
0

TESTS:

sage: SR(oo).arcsin()
Traceback (most recent call last):
...
RuntimeError: arcsin_eval(): arcsin(infinity) encountered
sage: SR(-oo).arcsin()
Traceback (most recent call last):
...
RuntimeError: arcsin_eval(): arcsin(infinity) encountered
sage: SR(unsigned_infinity).arcsin()
Infinity

arcsinh(hold=False)

Return the inverse hyperbolic sine of self.

EXAMPLES:

sage: x.arcsinh()
arcsinh(x)
sage: SR(0).arcsinh()
0
sage: SR(1).arcsinh()
arcsinh(1)
sage: SR(1.0).arcsinh()
0.881373587019543
sage: maxima('asinh(2.0)')
1.4436354751788...

Sage automatically applies certain identities:

sage: SR(3/2).arcsinh().cosh()
1/2*sqrt(13)

To prevent automatic evaluation use the hold argument:

sage: SR(-2).arcsinh()
-arcsinh(2)
sage: SR(-2).arcsinh(hold=True)
arcsinh(-2)

This also works using functional notation:

sage: arcsinh(-2,hold=True)
arcsinh(-2)
sage: arcsinh(-2)
-arcsinh(2)

To then evaluate again, we currently must use Maxima via simplify():

sage: a = SR(-2).arcsinh(hold=True); a.simplify()
-arcsinh(2)

TESTS:

sage: SR(oo).arcsinh()
+Infinity
sage: SR(-oo).arcsinh()
-Infinity
sage: SR(unsigned_infinity).arcsinh()
Infinity

arctan(hold=False)

Return the arc tangent of self.

EXAMPLES:

sage: x = var('x')
sage: x.arctan()
arctan(x)
sage: SR(1).arctan()
1/4*pi
sage: SR(1/2).arctan()
arctan(1/2)
sage: SR(0.5).arctan()
0.463647609000806
sage: plot(lambda x: SR(x).arctan(), -20,20)
Graphics object consisting of 1 graphics primitive

To prevent automatic evaluation use the hold argument:

sage: SR(1).arctan(hold=True)
arctan(1)

This also works using functional notation:

sage: arctan(1,hold=True)
arctan(1)
sage: arctan(1)
1/4*pi

To then evaluate again, we currently must use Maxima via simplify():

sage: a = SR(1).arctan(hold=True); a.simplify()
1/4*pi

TESTS:

sage: SR(oo).arctan()
1/2*pi
sage: SR(-oo).arctan()
-1/2*pi
sage: SR(unsigned_infinity).arctan()
Traceback (most recent call last):
...
RuntimeError: arctan_eval(): arctan(unsigned_infinity) encountered

arctan2(x, hold=False)

Return the inverse of the 2-variable tan function on self and x.

EXAMPLES:

sage: var('x,y')
(x, y)
sage: x.arctan2(y)
arctan2(x, y)
sage: SR(1/2).arctan2(1/2)
1/4*pi
sage: maxima.eval('atan2(1/2,1/2)')
'%pi/4'

sage: SR(-0.7).arctan2(SR(-0.6))
-2.27942259892257

To prevent automatic evaluation use the hold argument:

sage: SR(1/2).arctan2(1/2, hold=True)
arctan2(1/2, 1/2)

This also works using functional notation:

sage: arctan2(1,2,hold=True)
arctan2(1, 2)
sage: arctan2(1,2)
arctan(1/2)

To then evaluate again, we currently must use Maxima via simplify():

sage: a = SR(1/2).arctan2(1/2, hold=True); a.simplify()
1/4*pi

TESTS:

We compare a bunch of different evaluation points between Sage and Maxima:

sage: float(SR(0.7).arctan2(0.6))
0.8621700546672264
sage: maxima('atan2(0.7,0.6)')
0.8621700546672264
sage: float(SR(0.7).arctan2(-0.6))
2.279422598922567
sage: maxima('atan2(0.7,-0.6)')
2.279422598922567
sage: float(SR(-0.7).arctan2(0.6))
-0.8621700546672264
sage: maxima('atan2(-0.7,0.6)')
-0.8621700546672264
sage: float(SR(-0.7).arctan2(-0.6))
-2.279422598922567
sage: maxima('atan2(-0.7,-0.6)')
-2.279422598922567
sage: float(SR(0).arctan2(-0.6))
3.141592653589793
sage: maxima('atan2(0,-0.6)')
3.141592653589793
sage: float(SR(0).arctan2(0.6))
0.0
sage: maxima('atan2(0,0.6)')
0.0
sage: SR(0).arctan2(0) # see trac ticket #11423
Traceback (most recent call last):
...
RuntimeError: arctan2_eval(): arctan2(0,0) encountered
sage: SR(I).arctan2(1)
arctan2(I, 1)
sage: SR(CDF(0,1)).arctan2(1)
arctan2(1.0*I, 1)
sage: SR(1).arctan2(CDF(0,1))
arctan2(1, 1.0*I)

sage: arctan2(0,oo)
0
sage: SR(oo).arctan2(oo)
1/4*pi
sage: SR(oo).arctan2(0)
1/2*pi
sage: SR(-oo).arctan2(0)
-1/2*pi
sage: SR(-oo).arctan2(-2)
pi
sage: SR(unsigned_infinity).arctan2(2)
Traceback (most recent call last):
...
RuntimeError: arctan2_eval(): arctan2(x, unsigned_infinity) encountered
sage: SR(2).arctan2(oo)
1/2*pi
sage: SR(2).arctan2(-oo)
-1/2*pi
sage: SR(2).arctan2(SR(unsigned_infinity))
Traceback (most recent call last):
...
RuntimeError: arctan2_eval(): arctan2(unsigned_infinity, x) encountered

arctanh(hold=False)

Return the inverse hyperbolic tangent of self.

EXAMPLES:

sage: x.arctanh()
arctanh(x)
sage: SR(0).arctanh()
0
sage: SR(1/2).arctanh()
arctanh(1/2)
sage: SR(0.5).arctanh()
0.549306144334055
sage: SR(0.5).arctanh().tanh()
0.500000000000000
sage: maxima('atanh(0.5)')  # abs tol 2e-16
0.5493061443340548

To prevent automatic evaluation use the hold argument:

sage: SR(-1/2).arctanh()
-arctanh(1/2)
sage: SR(-1/2).arctanh(hold=True)
arctanh(-1/2)

This also works using functional notation:

sage: arctanh(-1/2,hold=True)
arctanh(-1/2)
sage: arctanh(-1/2)
-arctanh(1/2)

To then evaluate again, we currently must use Maxima via simplify():

sage: a = SR(-1/2).arctanh(hold=True); a.simplify()
-arctanh(1/2)

TESTS:

sage: SR(1).arctanh()
+Infinity
sage: SR(-1).arctanh()
-Infinity

sage: SR(oo).arctanh()
-1/2*I*pi
sage: SR(-oo).arctanh()
1/2*I*pi
sage: SR(unsigned_infinity).arctanh()
Traceback (most recent call last):
...
RuntimeError: arctanh_eval(): arctanh(unsigned_infinity) encountered

args()

EXAMPLES:

sage: x,y = var('x,y')
sage: f = x + y
sage: f.arguments()
(x, y)

sage: g = f.function(x)
sage: g.arguments()
(x,)

arguments()

EXAMPLES:

sage: x,y = var('x,y')
sage: f = x + y
sage: f.arguments()
(x, y)

sage: g = f.function(x)
sage: g.arguments()
(x,)

assume()

Assume that this equation holds. This is relevant for symbolic integration, among other things.

EXAMPLES: We call the assume method to assume that \(x>2\):

sage: (x > 2).assume()

Bool returns True below if the inequality is definitely known to be True.

sage: bool(x > 0)
True
sage: bool(x < 0)
False

This may or may not be True, so bool returns False:

sage: bool(x > 3)
False

If you make inconsistent or meaningless assumptions, Sage will let you know:

sage: forget()
sage: assume(x<0)
sage: assume(x>0)
Traceback (most recent call last):
...
ValueError: Assumption is inconsistent
sage: assumptions()
[x < 0]
sage: forget()

TESTS:

sage: v,c = var('v,c')
sage: assume(c != 0)
sage: integral((1+v^2/c^2)^3/(1-v^2/c^2)^(3/2),v)
83/8*v/sqrt(-v^2/c^2 + 1) - 17/8*v^3/(c^2*sqrt(-v^2/c^2 + 1)) - 1/4*v^5/(c^4*sqrt(-v^2/c^2 + 1)) - 75/8*arcsin(v/(c^2*sqrt(c^(-2))))/sqrt(c^(-2))
sage: forget()

binomial(k, hold=False)

Return binomial coefficient “self choose k”.

OUTPUT:

A symbolic expression.

EXAMPLES:

sage: var('x, y')
(x, y)
sage: SR(5).binomial(SR(3))
10
sage: x.binomial(SR(3))
1/6*(x - 1)*(x - 2)*x
sage: x.binomial(y)
binomial(x, y)

To prevent automatic evaluation use the hold argument:

sage: x.binomial(3, hold=True)
binomial(x, 3)
sage: SR(5).binomial(3, hold=True)
binomial(5, 3)

To then evaluate again, we currently must use Maxima via simplify():

sage: a = SR(5).binomial(3, hold=True); a.simplify()
10

The hold parameter is also supported in functional notation:

sage: binomial(5,3, hold=True)
binomial(5, 3)

TESTS:

Check if we handle zero correctly (trac ticket #8561):

sage: x.binomial(0)
1
sage: SR(0).binomial(0)
1

canonicalize_radical()

Choose a canonical branch of the given expression. The square root, cube root, natural log, etc. functions are multi-valued. The canonicalize_radical() method will choose one of these values based on a heuristic.

For example, sqrt(x^2) has two values: x, and -x. The canonicalize_radical() function will choose one of them, consistently, based on the behavior of the expression as x tends to positive infinity. The solution chosen is the one which exhibits this same behavior. Since sqrt(x^2) approaches positive infinity as x does, the solution chosen is x (which also tends to positive infinity).

Warning

As shown in the examples below, a canonical form is not always returned, i.e., two mathematically identical expressions might be converted to different expressions.

Assumptions are not taken into account during the transformation. This may result in a branch choice inconsistent with your assumptions.

ALGORITHM:

This uses the Maxima radcan() command. From the Maxima documentation:

Simplifies an expression, which can contain logs, exponentials, and radicals, by converting it into a form which is canonical over a large class of expressions and a given ordering of variables; that is, all functionally equivalent forms are mapped into a unique form. For a somewhat larger class of expressions, radcan produces a regular form. Two equivalent expressions in this class do not necessarily have the same appearance, but their difference can be simplified by radcan to zero.

For some expressions radcan is quite time consuming. This is the cost of exploring certain relationships among the components of the expression for simplifications based on factoring and partial fraction expansions of exponents.

EXAMPLES:

canonicalize_radical() can perform some of the same manipulations as log_expand():

sage: y = SR.symbol('y')
sage: f = log(x*y)
sage: f.log_expand()
log(x) + log(y)
sage: f.canonicalize_radical()
log(x) + log(y)

And also handles some exponential functions:

sage: f = (e^x-1)/(1+e^(x/2))
sage: f.canonicalize_radical()
e^(1/2*x) - 1

It can also be used to change the base of a logarithm when the arguments to log() are positive real numbers:

sage: f = log(8)/log(2)
sage: f.canonicalize_radical()
3

sage: a = SR.symbol('a')
sage: f = (log(x+x^2)-log(x))^a/log(1+x)^(a/2)
sage: f.canonicalize_radical()
log(x + 1)^(1/2*a)

The simplest example of counter-intuitive behavior is what happens when we take the square root of a square:

sage: sqrt(x^2).canonicalize_radical()
x

If you don’t want this kind of “simplification,” don’t use canonicalize_radical().

This behavior can also be triggered when the expression under the radical is not given explicitly as a square:

sage: sqrt(x^2 - 2*x + 1).canonicalize_radical()
x - 1

Another place where this can become confusing is with logarithms of complex numbers. Suppose x is complex with x == r*e^(I*t) (r real). Then log(x) is log(r) + I*(t + 2*k*pi) for some integer k.

Calling canonicalize_radical() will choose a branch, eliminating the solutions for all choices of k but one. Simplified by hand, the expression below is (1/2)*log(2) + I*pi*k for integer k. However, canonicalize_radical() will take each log expression, and choose one particular solution, dropping the other. When the results are subtracted, we’re left with no imaginary part:

sage: f = (1/2)*log(2*x) + (1/2)*log(1/x)
sage: f.canonicalize_radical()
1/2*log(2)

Naturally the result is wrong for some choices of x:

sage: f(x = -1)
I*pi + 1/2*log(2)

The example below shows two expressions e1 and e2 which are “simplified” to different expressions, while their difference is “simplified” to zero; thus canonicalize_radical() does not return a canonical form:

sage: e1 = 1/(sqrt(5)+sqrt(2))
sage: e2 = (sqrt(5)-sqrt(2))/3
sage: e1.canonicalize_radical()
1/(sqrt(5) + sqrt(2))
sage: e2.canonicalize_radical()
1/3*sqrt(5) - 1/3*sqrt(2)
sage: (e1-e2).canonicalize_radical()
0

The issue reported in trac ticket #3520 is a case where canonicalize_radical() causes a numerical integral to be calculated incorrectly:

sage: f1 = sqrt(25 - x) * sqrt( 1 + 1/(4*(25-x)) )
sage: f2 = f1.canonicalize_radical()
sage: numerical_integral(f1.real(), 0, 1)[0] # abs tol 1e-10
4.974852579915647
sage: numerical_integral(f2.real(), 0, 1)[0] # abs tol 1e-10
-4.974852579915647

TESTS:

This tests that trac ticket #11668 has been fixed (by trac ticket #12780):

sage: a,b = var('a b', domain='real')
sage: A = abs((a+I*b))^2
sage: A.canonicalize_radical()
a^2 + b^2
sage: imag(A)
0
sage: imag(A.canonicalize_radical())
0

Ensure that deprecation warnings are thrown for the old “simplify” aliases:

sage: x.simplify_radical()
doctest...: DeprecationWarning: simplify_radical is deprecated. Please use canonicalize_radical instead.
See http://trac.sagemath.org/11912 for details.
x
sage: x.radical_simplify()
doctest...: DeprecationWarning: radical_simplify is deprecated. Please use canonicalize_radical instead.
See http://trac.sagemath.org/11912 for details.
x
sage: x.simplify_exp()
doctest...: DeprecationWarning: simplify_exp is deprecated. Please use canonicalize_radical instead.
See http://trac.sagemath.org/11912 for details.
x
sage: x.exp_simplify()
doctest...: DeprecationWarning: exp_simplify is deprecated. Please use canonicalize_radical instead.
See http://trac.sagemath.org/11912 for details.
x

coeff(*args, **kwds)

Deprecated: Use coefficient() instead. See trac ticket #17438 for details.

coefficient(s, n=1)

Return the coefficient of \(s^n\) in this symbolic expression.

INPUT:

• s - expression
• n - integer, default 1

OUTPUT:

A symbolic expression. The coefficient of \(s^n\).

Sometimes it may be necessary to expand or factor first, since this is not done automatically.

EXAMPLES:

sage: var('x,y,a')
(x, y, a)
sage: f = 100 + a*x + x^3*sin(x*y) + x*y + x/y + 2*sin(x*y)/x; f
x^3*sin(x*y) + a*x + x*y + x/y + 2*sin(x*y)/x + 100
sage: f.collect(x)
x^3*sin(x*y) + (a + y + 1/y)*x + 2*sin(x*y)/x + 100
sage: f.coefficient(x,0)
100
sage: f.coefficient(x,-1)
2*sin(x*y)
sage: f.coefficient(x,1)
a + y + 1/y
sage: f.coefficient(x,2)
0
sage: f.coefficient(x,3)
sin(x*y)
sage: f.coefficient(x^3)
sin(x*y)
sage: f.coefficient(sin(x*y))
x^3 + 2/x
sage: f.collect(sin(x*y))
a*x + x*y + (x^3 + 2/x)*sin(x*y) + x/y + 100

sage: var('a, x, y, z')
(a, x, y, z)
sage: f = (a*sqrt(2))*x^2 + sin(y)*x^(1/2) + z^z
sage: f.coefficient(sin(y))
sqrt(x)
sage: f.coefficient(x^2)
sqrt(2)*a
sage: f.coefficient(x^(1/2))
sin(y)
sage: f.coefficient(1)
0
sage: f.coefficient(x, 0)
sqrt(x)*sin(y) + z^z

TESTS:

Check if trac ticket #9505 is fixed:

sage: var('x,y,z')
(x, y, z)
sage: f = x*y*z^2
sage: f.coefficient(x*y)
z^2
sage: f.coefficient(x*y, 2)
Traceback (most recent call last):
...
TypeError: n != 1 only allowed for s being a variable

Using coeff() is now deprecated (trac ticket #17438):

sage: x.coeff(x)
doctest:...: DeprecationWarning: coeff is deprecated. Please use coefficient instead.
See http://trac.sagemath.org/17438 for details.
1

coefficients(x=None, sparse=True)

Return the coefficients of this symbolic expression as a polynomial in x.

INPUT:

• x – optional variable.

OUTPUT:

Depending on the value of sparse,

• A list of pairs (expr, n), where expr is a symbolic expression and n is a power (sparse=True, default)
• A list of expressions where the n-th element is the coefficient of x^n when self is seen as polynomial in x (sparse=False).

EXAMPLES:

sage: var('x, y, a')
(x, y, a)
sage: p = x^3 - (x-3)*(x^2+x) + 1
sage: p.coefficients()
[[1, 0], [3, 1], [2, 2]]
sage: p.coefficients(sparse=False)
[1, 3, 2]
sage: p = x - x^3 + 5/7*x^5
sage: p.coefficients()
[[1, 1], [-1, 3], [5/7, 5]]
sage: p.coefficients(sparse=False)
[0, 1, 0, -1, 0, 5/7]
sage: p = expand((x-a*sqrt(2))^2 + x + 1); p
-2*sqrt(2)*a*x + 2*a^2 + x^2 + x + 1
sage: p.coefficients(a)
[[x^2 + x + 1, 0], [-2*sqrt(2)*x, 1], [2, 2]]
sage: p.coefficients(a, sparse=False)
[x^2 + x + 1, -2*sqrt(2)*x, 2]
sage: p.coefficients(x)
[[2*a^2 + 1, 0], [-2*sqrt(2)*a + 1, 1], [1, 2]]
sage: p.coefficients(x, sparse=False)
[2*a^2 + 1, -2*sqrt(2)*a + 1, 1]

TESTS:

The behaviour is undefined with noninteger or negative exponents:

sage: p = (17/3*a)*x^(3/2) + x*y + 1/x + x^x
sage: p.coefficients(x)
[[1, -1], [x^x, 0], [y, 1], [17/3*a, 3/2]]
sage: p.coefficients(x, sparse=False)
Traceback (most recent call last):
...
ValueError: Cannot return dense coefficient list with noninteger exponents.

Using coeffs() is now deprecated (trac ticket #17438):

sage: x.coeffs()
doctest:...: DeprecationWarning: coeffs is deprecated. Please use coefficients instead.
See http://trac.sagemath.org/17438 for details.
[[1, 1]]

Series coefficients are now handled correctly (trac ticket #17399):

sage: s=(1/(1-x)).series(x,6); s
1 + 1*x + 1*x^2 + 1*x^3 + 1*x^4 + 1*x^5 + Order(x^6)
sage: s.coefficients()
[[1, 0], [1, 1], [1, 2], [1, 3], [1, 4], [1, 5]]
sage: s.coefficients(x, sparse=False)
[1, 1, 1, 1, 1, 1]
sage: x,y = var("x,y")
sage: s=(1/(1-y*x-x)).series(x,3); s
1 + (y + 1)*x + ((y + 1)^2)*x^2 + Order(x^3)
sage: s.coefficients(x, sparse=False)
[1, y + 1, (y + 1)^2]

We can find coefficients of symbolic functions, trac ticket #12255:

sage: g = function('g', var('t'))
sage: f = 3*g + g**2 + t
sage: f.coefficients(g)
[[t, 0], [3, 1], [1, 2]]

coeffs(*args, **kwds)

Deprecated: Use coefficients() instead. See trac ticket #17438 for details.

collect(s)

Collect the coefficients of s into a group.

INPUT:

• s – the symbol whose coefficients will be collected.

OUTPUT:

A new expression, equivalent to the original one, with the coefficients of s grouped.

Note

The expression is not expanded or factored before the grouping takes place. For best results, call expand() on the expression before collect().

EXAMPLES:

In the first term of \(f\), \(x\) has a coefficient of \(4y\). In the second term, \(x\) has a coefficient of \(z\). Therefore, if we collect those coefficients, \(x\) will have a coefficient of \(4y+z\):

sage: x,y,z = var('x,y,z')
sage: f = 4*x*y + x*z + 20*y^2 + 21*y*z + 4*z^2 + x^2*y^2*z^2
sage: f.collect(x)
x^2*y^2*z^2 + x*(4*y + z) + 20*y^2 + 21*y*z + 4*z^2

Here we do the same thing for \(y\) and \(z\); however, note that we do not factor the \(y^{2}\) and \(z^{2}\) terms before collecting coefficients:

sage: f.collect(y)
(x^2*z^2 + 20)*y^2 + (4*x + 21*z)*y + x*z + 4*z^2
sage: f.collect(z)
(x^2*y^2 + 4)*z^2 + 4*x*y + 20*y^2 + (x + 21*y)*z

Sometimes, we do have to call expand() on the expression first to achieve the desired result:

sage: f = (x + y)*(x - z)
sage: f.collect(x)
x^2 + x*y - x*z - y*z
sage: f.expand().collect(x)
x^2 + x*(y - z) - y*z

TESTS:

The output should be equivalent to the input:

sage: polynomials = QQ['x']
sage: f = SR(polynomials.random_element())
sage: g = f.collect(x)
sage: bool(f == g)
True

If s is not present in the given expression, the expression should not be modified. The variable \(z\) will not be present in \(f\) below since \(f\) is a random polynomial of maximum degree 10 in \(x\) and \(y\):

sage: z = var('z')
sage: polynomials = QQ['x,y']
sage: f = SR(polynomials.random_element(10))
sage: g = f.collect(z)
sage: bool(str(f) == str(g))
True

Check if trac ticket #9046 is fixed:

sage: var('a b x y z')
(a, b, x, y, z)
sage: p = -a*x^3 - a*x*y^2 + 2*b*x^2*y + 2*y^3 + x^2*z + y^2*z + x^2 + y^2 + a*x
sage: p.collect(x)
-a*x^3 + (2*b*y + z + 1)*x^2 + 2*y^3 + y^2*z - (a*y^2 - a)*x + y^2

collect_common_factors()

This function does not perform a full factorization but only looks for factors which are already explicitly present.

Polynomials can often be brought into a more compact form by collecting common factors from the terms of sums. This is accomplished by this function.

EXAMPLES:

sage: var('x')
x
sage: (x/(x^2 + x)).collect_common_factors()
1/(x + 1)

sage: var('a,b,c,x,y')
(a, b, c, x, y)
sage: (a*x+a*y).collect_common_factors()
a*(x + y)
sage: (a*x^2+2*a*x*y+a*y^2).collect_common_factors()
(x^2 + 2*x*y + y^2)*a
sage: (a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y)).collect_common_factors()
((x + y)*y + x)*(a + c)*a*b

combine()

Return a simplified version of this symbolic expression by combining all terms with the same denominator into a single term.

EXAMPLES:

sage: var('x, y, a, b, c')
(x, y, a, b, c)
sage: f = x*(x-1)/(x^2 - 7) + y^2/(x^2-7) + 1/(x+1) + b/a + c/a; f
(x - 1)*x/(x^2 - 7) + y^2/(x^2 - 7) + b/a + c/a + 1/(x + 1)
sage: f.combine()
((x - 1)*x + y^2)/(x^2 - 7) + (b + c)/a + 1/(x + 1)

conjugate(hold=False)

Return the complex conjugate of this symbolic expression.

EXAMPLES:

sage: a = 1 + 2*I
sage: a.conjugate()
-2*I + 1
sage: a = sqrt(2) + 3^(1/3)*I; a
sqrt(2) + I*3^(1/3)
sage: a.conjugate()
sqrt(2) - I*3^(1/3)

sage: SR(CDF.0).conjugate()
-1.0*I
sage: x.conjugate()
conjugate(x)
sage: SR(RDF(1.5)).conjugate()
1.5
sage: SR(float(1.5)).conjugate()
1.5
sage: SR(I).conjugate()
-I
sage: ( 1+I  + (2-3*I)*x).conjugate()
(3*I + 2)*conjugate(x) - I + 1

Using the hold parameter it is possible to prevent automatic evaluation:

sage: SR(I).conjugate(hold=True)
conjugate(I)

This also works in functional notation:

sage: conjugate(I)
-I
sage: conjugate(I,hold=True)
conjugate(I)

To then evaluate again, we currently must use Maxima via simplify():

sage: a = SR(I).conjugate(hold=True); a.simplify()
-I

content(s)

Return the content of this expression when considered as a polynomial in s.

See also unit(), primitive_part(), and unit_content_primitive().

INPUT:

• s – a symbolic expression.

OUTPUT:

The content part of a polynomial as a symbolic expression. It is defined as the gcd of the coefficients.

Warning

The expression is considered to be a univariate polynomial in s. The output is different from the content() method provided by multivariate polynomial rings in Sage.

EXAMPLES:

sage: (2*x+4).content(x)
2
sage: (2*x+1).content(x)
1
sage: (2*x+1/2).content(x)
1/2
sage: var('y')
y
sage: (2*x + 4*sin(y)).content(sin(y))
2

contradicts(soln)

Return True if this relation is violated by the given variable assignment(s).

EXAMPLES:

sage: (x<3).contradicts(x==0)
False
sage: (x<3).contradicts(x==3)
True
sage: (x<=3).contradicts(x==3)
False
sage: y = var('y')
sage: (x<y).contradicts(x==30)
False
sage: (x<y).contradicts({x: 30, y: 20})
True

convert(target=None)

Call the convert function in the units package. For symbolic variables that are not units, this function just returns the variable.

INPUT:

• self – the symbolic expression converting from
• target – (default None) the symbolic expression converting to

OUTPUT:

A symbolic expression.

EXAMPLES:

sage: units.length.foot.convert()
381/1250*meter
sage: units.mass.kilogram.convert(units.mass.pound)
100000000/45359237*pound

We do not get anything new by converting an ordinary symbolic variable:

sage: a = var('a')
sage: a - a.convert()
0

Raises ValueError if self and target are not convertible:

sage: units.mass.kilogram.convert(units.length.foot)
Traceback (most recent call last):
...
ValueError: Incompatible units
sage: (units.length.meter^2).convert(units.length.foot)
Traceback (most recent call last):
...
ValueError: Incompatible units

Recognizes derived unit relationships to base units and other derived units:

sage: (units.length.foot/units.time.second^2).convert(units.acceleration.galileo)
762/25*galileo
sage: (units.mass.kilogram*units.length.meter/units.time.second^2).convert(units.force.newton)
newton
sage: (units.length.foot^3).convert(units.area.acre*units.length.inch)
1/3630*(acre*inch)
sage: (units.charge.coulomb).convert(units.current.ampere*units.time.second)
(ampere*second)
sage: (units.pressure.pascal*units.si_prefixes.kilo).convert(units.pressure.pounds_per_square_inch)
1290320000000/8896443230521*pounds_per_square_inch

For decimal answers multiply by 1.0:

sage: (units.pressure.pascal*units.si_prefixes.kilo).convert(units.pressure.pounds_per_square_inch)*1.0
0.145037737730209*pounds_per_square_inch

Converting temperatures works as well:

sage: s = 68*units.temperature.fahrenheit
sage: s.convert(units.temperature.celsius)
20*celsius
sage: s.convert()
293.150000000000*kelvin

Trying to multiply temperatures by another unit then converting raises a ValueError:

sage: wrong = 50*units.temperature.celsius*units.length.foot
sage: wrong.convert()
Traceback (most recent call last):
...
ValueError: Cannot convert

cos(hold=False)

Return the cosine of self.

EXAMPLES:

sage: var('x, y')
(x, y)
sage: cos(x^2 + y^2)
cos(x^2 + y^2)
sage: cos(sage.symbolic.constants.pi)
-1
sage: cos(SR(1))
cos(1)
sage: cos(SR(RealField(150)(1)))
0.54030230586813971740093660744297660373231042

In order to get a numeric approximation use .n():

sage: SR(RR(1)).cos().n()
0.540302305868140
sage: SR(float(1)).cos().n()
0.540302305868140

To prevent automatic evaluation use the hold argument:

sage: pi.cos()
-1
sage: pi.cos(hold=True)
cos(pi)

This also works using functional notation:

sage: cos(pi,hold=True)
cos(pi)
sage: cos(pi)
-1

To then evaluate again, we currently must use Maxima via simplify():

sage: a = pi.cos(hold=True); a.simplify()
-1

TESTS:

sage: SR(oo).cos()
Traceback (most recent call last):
...
RuntimeError: cos_eval(): cos(infinity) encountered
sage: SR(-oo).cos()
Traceback (most recent call last):
...
RuntimeError: cos_eval(): cos(infinity) encountered
sage: SR(unsigned_infinity).cos()
Traceback (most recent call last):
...
RuntimeError: cos_eval(): cos(infinity) encountered

cosh(hold=False)

Return cosh of self.

We have \(\cosh(x) = (e^{x} + e^{-x})/2\).

EXAMPLES:

sage: x.cosh()
cosh(x)
sage: SR(1).cosh()
cosh(1)
sage: SR(0).cosh()
1
sage: SR(1.0).cosh()
1.54308063481524
sage: maxima('cosh(1.0)')
1.54308063481524...
sage: SR(1.00000000000000000000000000).cosh()
1.5430806348152437784779056
sage: SR(RIF(1)).cosh()
1.543080634815244?

To prevent automatic evaluation use the hold argument:

sage: arcsinh(x).cosh()
sqrt(x^2 + 1)
sage: arcsinh(x).cosh(hold=True)
cosh(arcsinh(x))

This also works using functional notation:

sage: cosh(arcsinh(x),hold=True)
cosh(arcsinh(x))
sage: cosh(arcsinh(x))
sqrt(x^2 + 1)

To then evaluate again, we currently must use Maxima via simplify():

sage: a = arcsinh(x).cosh(hold=True); a.simplify()
sqrt(x^2 + 1)

TESTS:

sage: SR(oo).cosh()
+Infinity
sage: SR(-oo).cosh()
+Infinity
sage: SR(unsigned_infinity).cosh()
Traceback (most recent call last):
...
RuntimeError: cosh_eval(): cosh(unsigned_infinity) encountered

csgn(hold=False)

Return the sign of self, which is -1 if self < 0, 0 if self == 0, and 1 if self > 0, or unevaluated when self is a nonconstant symbolic expression.

If self is not real, return the complex half-plane (left or right) in which the number lies. If self is pure imaginary, return the sign of the imaginary part of self.

EXAMPLES:

sage: x = var('x')
sage: SR(-2).csgn()
-1
sage: SR(0.0).csgn()
0
sage: SR(10).csgn()
1
sage: x.csgn()
csgn(x)
sage: SR(CDF.0).csgn()
1
sage: SR(I).csgn()
1
sage: SR(-I).csgn()
-1
sage: SR(1+I).csgn()
1
sage: SR(1-I).csgn()
1
sage: SR(-1+I).csgn()
-1
sage: SR(-1-I).csgn()
-1

Using the hold parameter it is possible to prevent automatic evaluation:

sage: SR(I).csgn(hold=True)
csgn(I)

default_variable()

Return the default variable, which is by definition the first variable in self, or \(x\) is there are no variables in self. The result is cached.

EXAMPLES:

sage: sqrt(2).default_variable()
x
sage: x, theta, a = var('x, theta, a')
sage: f = x^2 + theta^3 - a^x
sage: f.default_variable()
a

Note that this is the first variable, not the first argument:

sage: f(theta, a, x) = a + theta^3
sage: f.default_variable()
a
sage: f.variables()
(a, theta)
sage: f.arguments()
(theta, a, x)

degree(s)

Return the exponent of the highest nonnegative power of s in self.

OUTPUT:

An integer >= 0.

EXAMPLES:

sage: var('x,y,a')
(x, y, a)
sage: f = 100 + a*x + x^3*sin(x*y) + x*y + x/y^10 + 2*sin(x*y)/x; f
x^3*sin(x*y) + a*x + x*y + 2*sin(x*y)/x + x/y^10 + 100
sage: f.degree(x)
3
sage: f.degree(y)
1
sage: f.degree(sin(x*y))
1
sage: (x^-3+y).degree(x)
0

denominator(normalize=True)

Return the denominator of this symbolic expression

INPUT:

• normalize – (default: True) a boolean.

If normalize is True, the expression is first normalized to have it as a fraction before getting the denominator.

If normalize is False, the expression is kept and if it is not a quotient, then this will just return 1.

See also

normalize(), numerator(), numerator_denominator(), combine()

EXAMPLES:

sage: x, y, z, theta = var('x, y, z, theta')
sage: f = (sqrt(x) + sqrt(y) + sqrt(z))/(x^10 - y^10 - sqrt(theta))
sage: f.numerator()
sqrt(x) + sqrt(y) + sqrt(z)
sage: f.denominator()
x^10 - y^10 - sqrt(theta)

sage: f.numerator(normalize=False)
(sqrt(x) + sqrt(y) + sqrt(z))
sage: f.denominator(normalize=False)
x^10 - y^10 - sqrt(theta)

sage: y = var('y')
sage: g = x + y/(x + 2); g
x + y/(x + 2)
sage: g.numerator(normalize=False)
x + y/(x + 2)
sage: g.denominator(normalize=False)
1

TESTS:

sage: ((x+y)^2/(x-y)^3*x^3).denominator(normalize=False)
(x - y)^3
sage: ((x+y)^2*x^3).denominator(normalize=False)
1
sage: (y/x^3).denominator(normalize=False)
x^3
sage: t = y/x^3/(x+y)^(1/2); t
y/(sqrt(x + y)*x^3)
sage: t.denominator(normalize=False)
sqrt(x + y)*x^3
sage: (1/x^3).denominator(normalize=False)
x^3
sage: (x^3).denominator(normalize=False)
1
sage: (y*x^sin(x)).denominator(normalize=False)
Traceback (most recent call last):
...
TypeError: self is not a rational expression

derivative(*args)

Return the derivative of this expressions with respect to the variables supplied in args.

Multiple variables and iteration counts may be supplied; see documentation for the global derivative() function for more details.

See also

This is implemented in the \(_derivative\) method (see the source code).

EXAMPLES:

sage: var("x y")
(x, y)
sage: t = (x^2+y)^2
sage: t.derivative(x)
4*(x^2 + y)*x
sage: t.derivative(x, 2)
12*x^2 + 4*y
sage: t.derivative(x, 2, y)
4
sage: t.derivative(y)
2*x^2 + 2*y

If the function depends on only one variable, you may omit the variable. Giving just a number (for the order of the derivative) also works:

sage: f(x) = x^3 + sin(x)
sage: f.derivative()
x |--> 3*x^2 + cos(x)
sage: f.derivative(2)
x |--> 6*x - sin(x)

sage: t = sin(x+y^2)*tan(x*y)
sage: t.derivative(x)
(tan(x*y)^2 + 1)*y*sin(y^2 + x) + cos(y^2 + x)*tan(x*y)
sage: t.derivative(y)
(tan(x*y)^2 + 1)*x*sin(y^2 + x) + 2*y*cos(y^2 + x)*tan(x*y)

sage: h = sin(x)/cos(x)
sage: derivative(h,x,x,x)
8*sin(x)^2/cos(x)^2 + 6*sin(x)^4/cos(x)^4 + 2
sage: derivative(h,x,3)
8*sin(x)^2/cos(x)^2 + 6*sin(x)^4/cos(x)^4 + 2

sage: var('x, y')
(x, y)
sage: u = (sin(x) + cos(y))*(cos(x) - sin(y))
sage: derivative(u,x,y)
-cos(x)*cos(y) + sin(x)*sin(y)
sage: f = ((x^2+1)/(x^2-1))^(1/4)
sage: g = derivative(f, x); g # this is a complex expression
-1/2*((x^2 + 1)*x/(x^2 - 1)^2 - x/(x^2 - 1))/((x^2 + 1)/(x^2 - 1))^(3/4)
sage: g.factor()
-x/((x + 1)^2*(x - 1)^2*((x^2 + 1)/(x^2 - 1))^(3/4))

sage: y = var('y')
sage: f = y^(sin(x))
sage: derivative(f, x)
y^sin(x)*cos(x)*log(y)

sage: g(x) = sqrt(5-2*x)
sage: g_3 = derivative(g, x, 3); g_3(2)
-3

sage: f = x*e^(-x)
sage: derivative(f, 100)
x*e^(-x) - 100*e^(-x)

sage: g = 1/(sqrt((x^2-1)*(x+5)^6))
sage: derivative(g, x)
-((x + 5)^6*x + 3*(x^2 - 1)*(x + 5)^5)/((x^2 - 1)*(x + 5)^6)^(3/2)

TESTS:

sage: t.derivative()
Traceback (most recent call last):
...
ValueError: No differentiation variable specified.

diff(*args)

Return the derivative of this expressions with respect to the variables supplied in args.

Multiple variables and iteration counts may be supplied; see documentation for the global derivative() function for more details.

See also

This is implemented in the \(_derivative\) method (see the source code).

EXAMPLES:

sage: var("x y")
(x, y)
sage: t = (x^2+y)^2
sage: t.derivative(x)
4*(x^2 + y)*x
sage: t.derivative(x, 2)
12*x^2 + 4*y
sage: t.derivative(x, 2, y)
4
sage: t.derivative(y)
2*x^2 + 2*y

If the function depends on only one variable, you may omit the variable. Giving just a number (for the order of the derivative) also works:

sage: f(x) = x^3 + sin(x)
sage: f.derivative()
x |--> 3*x^2 + cos(x)
sage: f.derivative(2)
x |--> 6*x - sin(x)

sage: t = sin(x+y^2)*tan(x*y)
sage: t.derivative(x)
(tan(x*y)^2 + 1)*y*sin(y^2 + x) + cos(y^2 + x)*tan(x*y)
sage: t.derivative(y)
(tan(x*y)^2 + 1)*x*sin(y^2 + x) + 2*y*cos(y^2 + x)*tan(x*y)

sage: h = sin(x)/cos(x)
sage: derivative(h,x,x,x)
8*sin(x)^2/cos(x)^2 + 6*sin(x)^4/cos(x)^4 + 2
sage: derivative(h,x,3)
8*sin(x)^2/cos(x)^2 + 6*sin(x)^4/cos(x)^4 + 2

sage: var('x, y')
(x, y)
sage: u = (sin(x) + cos(y))*(cos(x) - sin(y))
sage: derivative(u,x,y)
-cos(x)*cos(y) + sin(x)*sin(y)
sage: f = ((x^2+1)/(x^2-1))^(1/4)
sage: g = derivative(f, x); g # this is a complex expression
-1/2*((x^2 + 1)*x/(x^2 - 1)^2 - x/(x^2 - 1))/((x^2 + 1)/(x^2 - 1))^(3/4)
sage: g.factor()
-x/((x + 1)^2*(x - 1)^2*((x^2 + 1)/(x^2 - 1))^(3/4))

sage: y = var('y')
sage: f = y^(sin(x))
sage: derivative(f, x)
y^sin(x)*cos(x)*log(y)

sage: g(x) = sqrt(5-2*x)
sage: g_3 = derivative(g, x, 3); g_3(2)
-3

sage: f = x*e^(-x)
sage: derivative(f, 100)
x*e^(-x) - 100*e^(-x)

sage: g = 1/(sqrt((x^2-1)*(x+5)^6))
sage: derivative(g, x)
-((x + 5)^6*x + 3*(x^2 - 1)*(x + 5)^5)/((x^2 - 1)*(x + 5)^6)^(3/2)

TESTS:

sage: t.derivative()
Traceback (most recent call last):
...
ValueError: No differentiation variable specified.

differentiate(*args)

Return the derivative of this expressions with respect to the variables supplied in args.

Multiple variables and iteration counts may be supplied; see documentation for the global derivative() function for more details.

See also

This is implemented in the \(_derivative\) method (see the source code).

EXAMPLES:

sage: var("x y")
(x, y)
sage: t = (x^2+y)^2
sage: t.derivative(x)
4*(x^2 + y)*x
sage: t.derivative(x, 2)
12*x^2 + 4*y
sage: t.derivative(x, 2, y)
4
sage: t.derivative(y)
2*x^2 + 2*y

If the function depends on only one variable, you may omit the variable. Giving just a number (for the order of the derivative) also works:

sage: f(x) = x^3 + sin(x)
sage: f.derivative()
x |--> 3*x^2 + cos(x)
sage: f.derivative(2)
x |--> 6*x - sin(x)

sage: t = sin(x+y^2)*tan(x*y)
sage: t.derivative(x)
(tan(x*y)^2 + 1)*y*sin(y^2 + x) + cos(y^2 + x)*tan(x*y)
sage: t.derivative(y)
(tan(x*y)^2 + 1)*x*sin(y^2 + x) + 2*y*cos(y^2 + x)*tan(x*y)

sage: h = sin(x)/cos(x)
sage: derivative(h,x,x,x)
8*sin(x)^2/cos(x)^2 + 6*sin(x)^4/cos(x)^4 + 2
sage: derivative(h,x,3)
8*sin(x)^2/cos(x)^2 + 6*sin(x)^4/cos(x)^4 + 2

sage: var('x, y')
(x, y)
sage: u = (sin(x) + cos(y))*(cos(x) - sin(y))
sage: derivative(u,x,y)
-cos(x)*cos(y) + sin(x)*sin(y)
sage: f = ((x^2+1)/(x^2-1))^(1/4)
sage: g = derivative(f, x); g # this is a complex expression
-1/2*((x^2 + 1)*x/(x^2 - 1)^2 - x/(x^2 - 1))/((x^2 + 1)/(x^2 - 1))^(3/4)
sage: g.factor()
-x/((x + 1)^2*(x - 1)^2*((x^2 + 1)/(x^2 - 1))^(3/4))

sage: y = var('y')
sage: f = y^(sin(x))
sage: derivative(f, x)
y^sin(x)*cos(x)*log(y)

sage: g(x) = sqrt(5-2*x)
sage: g_3 = derivative(g, x, 3); g_3(2)
-3

sage: f = x*e^(-x)
sage: derivative(f, 100)
x*e^(-x) - 100*e^(-x)

sage: g = 1/(sqrt((x^2-1)*(x+5)^6))
sage: derivative(g, x)
-((x + 5)^6*x + 3*(x^2 - 1)*(x + 5)^5)/((x^2 - 1)*(x + 5)^6)^(3/2)

TESTS:

sage: t.derivative()
Traceback (most recent call last):
...
ValueError: No differentiation variable specified.

divide_both_sides(x, checksign=None)

Return a relation obtained by dividing both sides of this relation by x.

Note

The checksign keyword argument is currently ignored and is included for backward compatibility reasons only.

EXAMPLES:

sage: theta = var('theta')
sage: eqn =   (x^3 + theta < sin(x*theta))
sage: eqn.divide_both_sides(theta, checksign=False)
(x^3 + theta)/theta < sin(theta*x)/theta
sage: eqn.divide_both_sides(theta)
(x^3 + theta)/theta < sin(theta*x)/theta
sage: eqn/theta
(x^3 + theta)/theta < sin(theta*x)/theta

exp(hold=False)

Return exponential function of self, i.e., e to the power of self.

EXAMPLES:

sage: x.exp()
e^x
sage: SR(0).exp()
1
sage: SR(1/2).exp()
e^(1/2)
sage: SR(0.5).exp()
1.64872127070013
sage: math.exp(0.5)
1.6487212707001282

sage: SR(0.5).exp().log()
0.500000000000000
sage: (pi*I).exp()
-1

To prevent automatic evaluation use the hold argument:

sage: (pi*I).exp(hold=True)
e^(I*pi)

This also works using functional notation:

sage: exp(I*pi,hold=True)
e^(I*pi)
sage: exp(I*pi)
-1

To then evaluate again, we currently must use Maxima via simplify():

sage: a = (pi*I).exp(hold=True); a.simplify()
-1

TESTS:

Test if trac ticket #6377 is fixed:

sage: SR(oo).exp()
+Infinity
sage: SR(-oo).exp()
0
sage: SR(unsigned_infinity).exp()
Traceback (most recent call last):
...
RuntimeError: exp_eval(): exp^(unsigned_infinity) encountered

exp_simplify(*args, **kwds)

Deprecated: Use canonicalize_radical() instead. See trac ticket #11912 for details.

expand(side=None)

Expand this symbolic expression. Products of sums and exponentiated sums are multiplied out, numerators of rational expressions which are sums are split into their respective terms, and multiplications are distributed over addition at all levels.

EXAMPLES:

We expand the expression \((x-y)^5\) using both method and functional notation.

sage: x,y = var('x,y')
sage: a = (x-y)^5
sage: a.expand()
x^5 - 5*x^4*y + 10*x^3*y^2 - 10*x^2*y^3 + 5*x*y^4 - y^5
sage: expand(a)
x^5 - 5*x^4*y + 10*x^3*y^2 - 10*x^2*y^3 + 5*x*y^4 - y^5

We expand some other expressions:

sage: expand((x-1)^3/(y-1))
x^3/(y - 1) - 3*x^2/(y - 1) + 3*x/(y - 1) - 1/(y - 1)
sage: expand((x+sin((x+y)^2))^2)
x^2 + 2*x*sin((x + y)^2) + sin((x + y)^2)^2

We can expand individual sides of a relation:

sage: a = (16*x-13)^2 == (3*x+5)^2/2
sage: a.expand()
256*x^2 - 416*x + 169 == 9/2*x^2 + 15*x + 25/2
sage: a.expand('left')
256*x^2 - 416*x + 169 == 1/2*(3*x + 5)^2
sage: a.expand('right')
(16*x - 13)^2 == 9/2*x^2 + 15*x + 25/2

TESTS:

sage: var('x,y')
(x, y)
sage: ((x + (2/3)*y)^3).expand()
x^3 + 2*x^2*y + 4/3*x*y^2 + 8/27*y^3
sage: expand( (x*sin(x) - cos(y)/x)^2 )
x^2*sin(x)^2 - 2*cos(y)*sin(x) + cos(y)^2/x^2
sage: f = (x-y)*(x+y); f
(x + y)*(x - y)
sage: f.expand()
x^2 - y^2

sage: a,b,c = var('a,b,c')
sage: x,y = var('x,y', domain='real')
sage: p,q = var('p,q', domain='positive')
sage: (c/2*(5*(3*a*b*x*y*p*q)^2)^(7/2*c)).expand()
1/2*45^(7/2*c)*(a^2*b^2*x^2*y^2)^(7/2*c)*c*p^(7*c)*q^(7*c)
sage: ((-(-a*x*p)^3*(b*y*p)^3)^(c/2)).expand()
(a^3*b^3*x^3*y^3)^(1/2*c)*p^(3*c)
sage: x,y,p,q = var('x,y,p,q', domain='complex')

expand_log(algorithm='products')

Simplify symbolic expression, which can contain logs.

Expands logarithms of powers, logarithms of products and logarithms of quotients. The option algorithm specifies which expression types should be expanded.

INPUT:

• self - expression to be simplified

• algorithm - (default: ‘products’) optional, governs which expression is expanded. Possible values are

  • ‘nothing’ (no expansion),
  • ‘powers’ (log(a^r) is expanded),
  • ‘products’ (like ‘powers’ and also log(a*b) are expanded),
  • ‘all’ (all possible expansion).

  See also examples below.

DETAILS: This uses the Maxima simplifier and sets logexpand option for this simplifier. From the Maxima documentation: “Logexpand:true causes log(a^b) to become b*log(a). If it is set to all, log(a*b) will also simplify to log(a)+log(b). If it is set to super, then log(a/b) will also simplify to log(a)-log(b) for rational numbers a/b, a#1. (log(1/b), for integer b, always simplifies.) If it is set to false, all of these simplifications will be turned off. “

ALIAS: log_expand() and expand_log() are the same

EXAMPLES:

By default powers and products (and quotients) are expanded, but not quotients of integers:

sage: (log(3/4*x^pi)).log_expand()
pi*log(x) + log(3/4)

To expand also log(3/4) use algorithm='all':

sage: (log(3/4*x^pi)).log_expand('all')
pi*log(x) - log(4) + log(3)

To expand only the power use algorithm='powers'.:

sage: (log(x^6)).log_expand('powers')
6*log(x)

The expression log((3*x)^6) is not expanded with algorithm='powers', since it is converted into product first:

sage: (log((3*x)^6)).log_expand('powers')
log(729*x^6)

This shows that the option algorithm from the previous call has no influence to future calls (we changed some default Maxima flag, and have to ensure that this flag has been restored):

sage: (log(3/4*x^pi)).log_expand()
pi*log(x) + log(3/4)

sage: (log(3/4*x^pi)).log_expand('all')
pi*log(x) - log(4) + log(3)

sage: (log(3/4*x^pi)).log_expand()
pi*log(x) + log(3/4)

TESTS:

Most of these log expansions only make sense over the reals. So, we should set the Maxima domain variable to ‘real’ before we call out to Maxima. When we return, however, we should set the domain back to what it was, rather than assuming that it was ‘complex’. See trac ticket #12780:

sage: from sage.calculus.calculus import maxima
sage: maxima('domain: real;')
real
sage: x.expand_log()
x
sage: maxima('domain;')
real
sage: maxima('domain: complex;')
complex

AUTHORS:

• Robert Marik (11-2009)

expand_rational(side=None)

Expand this symbolic expression. Products of sums and exponentiated sums are multiplied out, numerators of rational expressions which are sums are split into their respective terms, and multiplications are distributed over addition at all levels.

EXAMPLES:

We expand the expression \((x-y)^5\) using both method and functional notation.

sage: x,y = var('x,y')
sage: a = (x-y)^5
sage: a.expand()
x^5 - 5*x^4*y + 10*x^3*y^2 - 10*x^2*y^3 + 5*x*y^4 - y^5
sage: expand(a)
x^5 - 5*x^4*y + 10*x^3*y^2 - 10*x^2*y^3 + 5*x*y^4 - y^5

We expand some other expressions:

sage: expand((x-1)^3/(y-1))
x^3/(y - 1) - 3*x^2/(y - 1) + 3*x/(y - 1) - 1/(y - 1)
sage: expand((x+sin((x+y)^2))^2)
x^2 + 2*x*sin((x + y)^2) + sin((x + y)^2)^2

We can expand individual sides of a relation:

sage: a = (16*x-13)^2 == (3*x+5)^2/2
sage: a.expand()
256*x^2 - 416*x + 169 == 9/2*x^2 + 15*x + 25/2
sage: a.expand('left')
256*x^2 - 416*x + 169 == 1/2*(3*x + 5)^2
sage: a.expand('right')
(16*x - 13)^2 == 9/2*x^2 + 15*x + 25/2

TESTS:

sage: var('x,y')
(x, y)
sage: ((x + (2/3)*y)^3).expand()
x^3 + 2*x^2*y + 4/3*x*y^2 + 8/27*y^3
sage: expand( (x*sin(x) - cos(y)/x)^2 )
x^2*sin(x)^2 - 2*cos(y)*sin(x) + cos(y)^2/x^2
sage: f = (x-y)*(x+y); f
(x + y)*(x - y)
sage: f.expand()
x^2 - y^2

sage: a,b,c = var('a,b,c')
sage: x,y = var('x,y', domain='real')
sage: p,q = var('p,q', domain='positive')
sage: (c/2*(5*(3*a*b*x*y*p*q)^2)^(7/2*c)).expand()
1/2*45^(7/2*c)*(a^2*b^2*x^2*y^2)^(7/2*c)*c*p^(7*c)*q^(7*c)
sage: ((-(-a*x*p)^3*(b*y*p)^3)^(c/2)).expand()
(a^3*b^3*x^3*y^3)^(1/2*c)*p^(3*c)
sage: x,y,p,q = var('x,y,p,q', domain='complex')

expand_sum()

For every symbolic sum in the given expression, try to expand it, symbolically or numerically.

While symbolic sum expressions with constant limits are evaluated immediately on the command line, unevaluated sums of this kind can result from, e.g., substitution of limit variables.

INPUT:

• self - symbolic expression

EXAMPLES:

sage: (k,n) = var('k,n')
sage: ex = sum(abs(-k*k+n),k,1,n)(n=8); ex
sum(abs(-k^2 + 8), k, 1, 8)
sage: ex.expand_sum()
162
sage: f(x,k) = sum((2/n)*(sin(n*x)*(-1)^(n+1)), n, 1, k)
sage: f(x,2)
-2*sum((-1)^n*sin(n*x)/n, n, 1, 2)
sage: f(x,2).expand_sum()
-sin(2*x) + 2*sin(x)

We can use this to do floating-point approximation as well:

sage: (k,n) = var('k,n')
sage: f(n)=sum(sqrt(abs(-k*k+n)),k,1,n)
sage: f(n=8)
sum(sqrt(abs(-k^2 + 8)), k, 1, 8)
sage: f(8).expand_sum()
sqrt(41) + sqrt(17) + 2*sqrt(14) + 3*sqrt(7) + 2*sqrt(2) + 3
sage: f(8).expand_sum().n()
31.7752256945384

See trac ticket #9424 for making the following no longer raise an error:

sage: f(8).n()
Traceback (most recent call last):
...
TypeError: cannot evaluate symbolic expression numerically

expand_trig(full=False, half_angles=False, plus=True, times=True)

Expand trigonometric and hyperbolic functions of sums of angles and of multiple angles occurring in self. For best results, self should already be expanded.

INPUT:

• full - (default: False) To enhance user control of simplification, this function expands only one level at a time by default, expanding sums of angles or multiple angles. To obtain full expansion into sines and cosines immediately, set the optional parameter full to True.
• half_angles - (default: False) If True, causes half-angles to be simplified away.
• plus - (default: True) Controls the sum rule; expansion of sums (e.g. ‘sin(x + y)’) will take place only if plus is True.
• times - (default: True) Controls the product rule, expansion of products (e.g. sin(2*x)) will take place only if times is True.

OUTPUT:

A symbolic expression.

EXAMPLES:

sage: sin(5*x).expand_trig()
5*cos(x)^4*sin(x) - 10*cos(x)^2*sin(x)^3 + sin(x)^5
sage: cos(2*x + var('y')).expand_trig()
cos(2*x)*cos(y) - sin(2*x)*sin(y)

We illustrate various options to this function:

sage: f = sin(sin(3*cos(2*x))*x)
sage: f.expand_trig()
sin((3*cos(cos(2*x))^2*sin(cos(2*x)) - sin(cos(2*x))^3)*x)
sage: f.expand_trig(full=True)
sin((3*(cos(cos(x)^2)*cos(sin(x)^2) + sin(cos(x)^2)*sin(sin(x)^2))^2*(cos(sin(x)^2)*sin(cos(x)^2) - cos(cos(x)^2)*sin(sin(x)^2)) - (cos(sin(x)^2)*sin(cos(x)^2) - cos(cos(x)^2)*sin(sin(x)^2))^3)*x)
sage: sin(2*x).expand_trig(times=False)
sin(2*x)
sage: sin(2*x).expand_trig(times=True)
2*cos(x)*sin(x)
sage: sin(2 + x).expand_trig(plus=False)
sin(x + 2)
sage: sin(2 + x).expand_trig(plus=True)
cos(x)*sin(2) + cos(2)*sin(x)
sage: sin(x/2).expand_trig(half_angles=False)
sin(1/2*x)
sage: sin(x/2).expand_trig(half_angles=True)
(-1)^floor(1/2*x/pi)*sqrt(-1/2*cos(x) + 1/2)

ALIASES:

trig_expand() and expand_trig() are the same

factor(dontfactor=[])

Factor the expression, containing any number of variables or functions, into factors irreducible over the integers.

INPUT:

• self - a symbolic expression
• dontfactor - list (default: []), a list of variables with respect to which factoring is not to occur. Factoring also will not take place with respect to any variables which are less important (using the variable ordering assumed for CRE form) than those on the ‘dontfactor’ list.

EXAMPLES:

sage: x,y,z = var('x, y, z')
sage: (x^3-y^3).factor()
(x^2 + x*y + y^2)*(x - y)
sage: factor(-8*y - 4*x + z^2*(2*y + x))
(x + 2*y)*(z + 2)*(z - 2)
sage: f = -1 - 2*x - x^2 + y^2 + 2*x*y^2 + x^2*y^2
sage: F = factor(f/(36*(1 + 2*y + y^2)), dontfactor=[x]); F
1/36*(x^2 + 2*x + 1)*(y - 1)/(y + 1)

If you are factoring a polynomial with rational coefficients (and dontfactor is empty) the factorization is done using Singular instead of Maxima, so the following is very fast instead of dreadfully slow:

sage: var('x,y')
(x, y)
sage: (x^99 + y^99).factor()
(x^60 + x^57*y^3 - x^51*y^9 - x^48*y^12 + x^42*y^18 + x^39*y^21 -
x^33*y^27 - x^30*y^30 - x^27*y^33 + x^21*y^39 + x^18*y^42 -
x^12*y^48 - x^9*y^51 + x^3*y^57 + y^60)*(x^20 + x^19*y -
x^17*y^3 - x^16*y^4 + x^14*y^6 + x^13*y^7 - x^11*y^9 -
x^10*y^10 - x^9*y^11 + x^7*y^13 + x^6*y^14 - x^4*y^16 -
x^3*y^17 + x*y^19 + y^20)*(x^10 - x^9*y + x^8*y^2 - x^7*y^3 +
x^6*y^4 - x^5*y^5 + x^4*y^6 - x^3*y^7 + x^2*y^8 - x*y^9 +
y^10)*(x^6 - x^3*y^3 + y^6)*(x^2 - x*y + y^2)*(x + y)

factor_list(dontfactor=[])

Return a list of the factors of self, as computed by the factor command.

INPUT:

• self - a symbolic expression
• dontfactor - see docs for factor()

Note

If you already have a factored expression and just want to get at the individual factors, use the \(_factor_list\) method instead.

EXAMPLES:

sage: var('x, y, z')
(x, y, z)
sage: f = x^3-y^3
sage: f.factor()
(x^2 + x*y + y^2)*(x - y)

Notice that the -1 factor is separated out:

sage: f.factor_list()
[(x^2 + x*y + y^2, 1), (x - y, 1)]

We factor a fairly straightforward expression:

sage: factor(-8*y - 4*x + z^2*(2*y + x)).factor_list()
[(x + 2*y, 1), (z + 2, 1), (z - 2, 1)]

A more complicated example:

sage: var('x, u, v')
(x, u, v)
sage: f = expand((2*u*v^2-v^2-4*u^3)^2 * (-u)^3 * (x-sin(x))^3)
sage: f.factor()
-(4*u^3 - 2*u*v^2 + v^2)^2*u^3*(x - sin(x))^3
sage: g = f.factor_list(); g
[(4*u^3 - 2*u*v^2 + v^2, 2), (u, 3), (x - sin(x), 3), (-1, 1)]

This function also works for quotients:

sage: f = -1 - 2*x - x^2 + y^2 + 2*x*y^2 + x^2*y^2
sage: g = f/(36*(1 + 2*y + y^2)); g
1/36*(x^2*y^2 + 2*x*y^2 - x^2 + y^2 - 2*x - 1)/(y^2 + 2*y + 1)
sage: g.factor(dontfactor=[x])
1/36*(x^2 + 2*x + 1)*(y - 1)/(y + 1)
sage: g.factor_list(dontfactor=[x])
[(x^2 + 2*x + 1, 1), (y + 1, -1), (y - 1, 1), (1/36, 1)]

This example also illustrates that the exponents do not have to be integers:

sage: f = x^(2*sin(x)) * (x-1)^(sqrt(2)*x); f
(x - 1)^(sqrt(2)*x)*x^(2*sin(x))
sage: f.factor_list()
[(x - 1, sqrt(2)*x), (x, 2*sin(x))]

factorial(hold=False)

Return the factorial of self.

OUTPUT:

A symbolic expression.

EXAMPLES:

sage: var('x, y')
(x, y)
sage: SR(5).factorial()
120
sage: x.factorial()
factorial(x)
sage: (x^2+y^3).factorial()
factorial(y^3 + x^2)

To prevent automatic evaluation use the hold argument:

sage: SR(5).factorial(hold=True)
factorial(5)

This also works using functional notation:

sage: factorial(5,hold=True)
factorial(5)
sage: factorial(5)
120

To then evaluate again, we currently must use Maxima via simplify():

sage: a = SR(5).factorial(hold=True); a.simplify()
120

factorial_simplify()

Simplify by combining expressions with factorials, and by expanding binomials into factorials.

ALIAS: factorial_simplify and simplify_factorial are the same

EXAMPLES:

Some examples are relatively clear:

sage: var('n,k')
(n, k)
sage: f = factorial(n+1)/factorial(n); f
factorial(n + 1)/factorial(n)
sage: f.simplify_factorial()
n + 1

sage: f = factorial(n)*(n+1); f
(n + 1)*factorial(n)
sage: simplify(f)
(n + 1)*factorial(n)
sage: f.simplify_factorial()
factorial(n + 1)

sage: f = binomial(n, k)*factorial(k)*factorial(n-k); f
binomial(n, k)*factorial(k)*factorial(-k + n)
sage: f.simplify_factorial()
factorial(n)

A more complicated example, which needs further processing:

sage: f = factorial(x)/factorial(x-2)/2 + factorial(x+1)/factorial(x)/2; f
1/2*factorial(x + 1)/factorial(x) + 1/2*factorial(x)/factorial(x - 2)
sage: g = f.simplify_factorial(); g
1/2*(x - 1)*x + 1/2*x + 1/2
sage: g.simplify_rational()
1/2*x^2 + 1/2

TESTS:

Check that the problem with applying \(full_simplify()\) to gamma functions (trac ticket #9240) has been fixed:

sage: gamma(1/3)
gamma(1/3)
sage: gamma(1/3).full_simplify()
gamma(1/3)
sage: gamma(4/3)
gamma(4/3)
sage: gamma(4/3).full_simplify()
1/3*gamma(1/3)

find(pattern)

Find all occurrences of the given pattern in this expression.

Note that once a subexpression matches the pattern, the search does not extend to subexpressions of it.

EXAMPLES:

sage: var('x,y,z,a,b')
(x, y, z, a, b)
sage: w0 = SR.wild(0); w1 = SR.wild(1)

sage: (sin(x)*sin(y)).find(sin(w0))
[sin(y), sin(x)]

sage: ((sin(x)+sin(y))*(a+b)).expand().find(sin(w0))
[sin(y), sin(x)]

sage: (1+x+x^2+x^3).find(x)
[x]
sage: (1+x+x^2+x^3).find(x^w0)
[x^2, x^3]

sage: (1+x+x^2+x^3).find(y)
[]

# subexpressions of a match are not listed
sage: ((x^y)^z).find(w0^w1)
[(x^y)^z]

find_local_maximum(a, b, var=None, tol=1.48e-08, maxfun=500)

Numerically find a local maximum of the expression self on the interval [a,b] (or [b,a]) along with the point at which the maximum is attained.

See the documentation for find_local_minimum() for more details.

EXAMPLES:

sage: f = x*cos(x)
sage: f.find_local_maximum(0,5)
(0.5610963381910451, 0.8603335890...)
sage: f.find_local_maximum(0,5, tol=0.1, maxfun=10)
(0.561090323458081..., 0.857926501456...)

find_local_minimum(a, b, var=None, tol=1.48e-08, maxfun=500)

Numerically find a local minimum of the expression self on the interval [a,b] (or [b,a]) and the point at which it attains that minimum. Note that self must be a function of (at most) one variable.

INPUT:

• var - variable (default: first variable in self)
• a,b - endpoints of interval on which to minimize self.
• tol - the convergence tolerance
• maxfun - maximum function evaluations

OUTPUT:

A tuple (minval, x), where

• minval – float. The minimum value that self takes on in the interval [a,b].
• x – float. The point at which self takes on the minimum value.

EXAMPLES:

sage: f = x*cos(x)
sage: f.find_local_minimum(1, 5)
(-3.288371395590..., 3.4256184695...)
sage: f.find_local_minimum(1, 5, tol=1e-3)
(-3.288371361890..., 3.4257507903...)
sage: f.find_local_minimum(1, 5, tol=1e-2, maxfun=10)
(-3.288370845983..., 3.4250840220...)
sage: show(f.plot(0, 20))
sage: f.find_local_minimum(1, 15)
(-9.477294259479..., 9.5293344109...)

ALGORITHM:

Uses sage.numerical.optimize.find_local_minimum().

AUTHORS:

• William Stein (2007-12-07)

find_root(a, b, var=None, xtol=1e-12, rtol=4.5e-16, maxiter=100, full_output=False)

Numerically find a root of self on the closed interval [a,b] (or [b,a]) if possible, where self is a function in the one variable. Note: this function only works in fixed (machine) precision, it is not possible to get arbitrary precision approximations with it.

INPUT:

• a, b - endpoints of the interval
• var - optional variable
• xtol, rtol - the routine converges when a root is known to lie within xtol of the value return. Should be >= 0. The routine modifies this to take into account the relative precision of doubles.
• maxiter - integer; if convergence is not achieved in maxiter iterations, an error is raised. Must be >= 0.
• full_output - bool (default: False), if True, also return object that contains information about convergence.

EXAMPLES:

Note that in this example both f(-2) and f(3) are positive, yet we still find a root in that interval:

sage: f = x^2 - 1
sage: f.find_root(-2, 3)
1.0
sage: f.find_root(-2, 3, x)
1.0
sage: z, result = f.find_root(-2, 3, full_output=True)
sage: result.converged
True
sage: result.flag
'converged'
sage: result.function_calls
11
sage: result.iterations
10
sage: result.root
1.0

More examples:

sage: (sin(x) + exp(x)).find_root(-10, 10)
-0.588532743981862...
sage: sin(x).find_root(-1,1)
0.0
sage: (1/tan(x)).find_root(3,3.5)
3.1415926535...

An example with a square root:

sage: f = 1 + x + sqrt(x+2); f.find_root(-2,10)
-1.618033988749895

Some examples that Ted Kosan came up with:

sage: t = var('t')
sage: v = 0.004*(9600*e^(-(1200*t)) - 2400*e^(-(300*t)))
sage: v.find_root(0, 0.002)
0.001540327067911417...

With this expression, we can see there is a zero very close to the origin:

sage: a = .004*(8*e^(-(300*t)) - 8*e^(-(1200*t)))*(720000*e^(-(300*t)) - 11520000*e^(-(1200*t))) +.004*(9600*e^(-(1200*t)) - 2400*e^(-(300*t)))^2
sage: show(plot(a, 0, .002), xmin=0, xmax=.002)

It is easy to approximate with find_root:

sage: a.find_root(0,0.002)
0.0004110514049349...

Using solve takes more effort, and even then gives only a solution with free (integer) variables:

sage: a.solve(t)
[]
sage: b = a.canonicalize_radical(); b
-23040.0*(-2.0*e^(1800*t) + 25.0*e^(900*t) - 32.0)*e^(-2400*t)
sage: b.solve(t)
[]
sage: b.solve(t, to_poly_solve=True)
[t == 1/450*I*pi*z... + 1/900*log(-3/4*sqrt(41) + 25/4),
 t == 1/450*I*pi*z... + 1/900*log(3/4*sqrt(41) + 25/4)]
sage: n(1/900*log(-3/4*sqrt(41) + 25/4))
0.000411051404934985

We illustrate that root finding is only implemented in one dimension:

sage: x, y = var('x,y')
sage: (x-y).find_root(-2,2)
Traceback (most recent call last):
...
NotImplementedError: root finding currently only implemented in 1 dimension.

TESTS:

Test the special case that failed for the first attempt to fix trac ticket #3980:

sage: t = var('t')
sage: find_root(1/t - x,0,2)
Traceback (most recent call last):
...
NotImplementedError: root finding currently only implemented in 1 dimension.

forget()

Forget the given constraint.

EXAMPLES:

sage: var('x,y')
(x, y)
sage: forget()
sage: assume(x>0, y < 2)
sage: assumptions()
[x > 0, y < 2]
sage: forget(y < 2)
sage: assumptions()
[x > 0]

TESTS:

Check if trac ticket #7507 is fixed:

sage: forget()
sage: n = var('n')
sage: foo=sin((-1)*n*pi)
sage: foo.simplify()
-sin(pi*n)
sage: assume(n, 'odd')
sage: assumptions()
[n is odd]
sage: foo.simplify()
0
sage: forget(n, 'odd')
sage: assumptions()
[]
sage: foo.simplify()
-sin(pi*n)

fraction(base_ring)

Return this expression as element of the algebraic fraction field over the base ring given.

EXAMPLES:

sage: fr = (1/x).fraction(ZZ); fr
1/x
sage: parent(fr)
Fraction Field of Univariate Polynomial Ring in x over Integer Ring
sage: parent(((pi+sqrt(2)/x).fraction(SR)))
Fraction Field of Univariate Polynomial Ring in x over Symbolic Ring
sage: parent(((pi+sqrt(2))/x).fraction(SR))
Fraction Field of Univariate Polynomial Ring in x over Symbolic Ring
sage: y=var('y')
sage: fr=((3*x^5 - 5*y^5)^7/(x*y)).fraction(GF(7)); fr
(3*x^35 + 2*y^35)/(x*y)
sage: parent(fr)
Fraction Field of Multivariate Polynomial Ring in x, y over Finite Field of size 7

TESTS:

Check that trac ticket #17736 is fixed:

sage: a,b,c = var('a,b,c')
sage: fr = (1/a).fraction(QQ); fr
1/a
sage: parent(fr)
Fraction Field of Univariate Polynomial Ring in a over Rational Field
sage: parent((b/(a+sin(c))).fraction(SR))
Fraction Field of Multivariate Polynomial Ring in a, b over Symbolic Ring

full_simplify()

Apply simplify_factorial(), simplify_rectform(), simplify_trig(), simplify_rational(), and then expand_sum() to self (in that order).

ALIAS: simplify_full and full_simplify are the same.

EXAMPLES:

sage: f = sin(x)^2 + cos(x)^2
sage: f.simplify_full()
1

sage: f = sin(x/(x^2 + x))
sage: f.simplify_full()
sin(1/(x + 1))

sage: var('n,k')
(n, k)
sage: f = binomial(n,k)*factorial(k)*factorial(n-k)
sage: f.simplify_full()
factorial(n)

TESTS:

There are two square roots of \($(x + 1)^2$\), so this should not be simplified to \($x + 1$\), trac ticket #12737:

sage: f = sqrt((x + 1)^2)
sage: f.simplify_full()
sqrt(x^2 + 2*x + 1)

The imaginary part of an expression should not change under simplification; trac ticket #11934:

sage: f = sqrt(-8*(4*sqrt(2) - 7)*x^4 + 16*(3*sqrt(2) - 5)*x^3)
sage: original = f.imag_part()
sage: simplified = f.full_simplify().imag_part()
sage: original - simplified
0

The invalid simplification from trac ticket #12322 should not occur after trac ticket #12737:

sage: t = var('t')
sage: assume(t, 'complex')
sage: assumptions()
[t is complex]
sage: f = (1/2)*log(2*t) + (1/2)*log(1/t)
sage: f.simplify_full()
1/2*log(2*t) - 1/2*log(t)
sage: forget()

Complex logs are not contracted, trac ticket #17556:

sage: x,y = SR.var('x,y')
sage: assume(y, 'complex')
sage: f = log(x*y) - (log(x) + log(y))
sage: f.simplify_full()
log(x*y) - log(x) - log(y)
sage: forget()

The simplifications from simplify_rectform() are performed, trac ticket #17556:

sage: f = ( e^(I*x) - e^(-I*x) ) / ( I*e^(I*x) + I*e^(-I*x) )
sage: f.simplify_full()
sin(x)/cos(x)

function(*args)

Return a callable symbolic expression with the given variables.

EXAMPLES:

We will use several symbolic variables in the examples below:

sage: var('x, y, z, t, a, w, n')
(x, y, z, t, a, w, n)

sage: u = sin(x) + x*cos(y)
sage: g = u.function(x,y)
sage: g(x,y)
x*cos(y) + sin(x)
sage: g(t,z)
t*cos(z) + sin(t)
sage: g(x^2, x^y)
x^2*cos(x^y) + sin(x^2)

sage: f = (x^2 + sin(a*w)).function(a,x,w); f
(a, x, w) |--> x^2 + sin(a*w)
sage: f(1,2,3)
sin(3) + 4

Using the function() method we can obtain the above function \(f\), but viewed as a function of different variables:

sage: h = f.function(w,a); h
(w, a) |--> x^2 + sin(a*w)

This notation also works:

sage: h(w,a) = f
sage: h
(w, a) |--> x^2 + sin(a*w)

You can even make a symbolic expression \(f\) into a function by writing f(x,y) = f:

sage: f = x^n + y^n; f
x^n + y^n
sage: f(x,y) = f
sage: f
(x, y) |--> x^n + y^n
sage: f(2,3)
3^n + 2^n

gamma(hold=False)

Return the Gamma function evaluated at self.

EXAMPLES:

sage: x = var('x')
sage: x.gamma()
gamma(x)
sage: SR(2).gamma()
1
sage: SR(10).gamma()
362880
sage: SR(10.0r).gamma()  # For ARM: rel tol 2e-15
362880.0
sage: SR(CDF(1,1)).gamma()
0.49801566811835607 - 0.15494982830181067*I

sage: gp('gamma(1+I)')
0.4980156681183560427136911175 - 0.1549498283018106851249551305*I # 32-bit
0.49801566811835604271369111746219809195 - 0.15494982830181068512495513048388660520*I # 64-bit

We plot the familiar plot of this log-convex function:

sage: plot(gamma(x), -6,4).show(ymin=-3,ymax=3)

To prevent automatic evaluation use the hold argument:

sage: SR(1/2).gamma()
sqrt(pi)
sage: SR(1/2).gamma(hold=True)
gamma(1/2)

This also works using functional notation:

sage: gamma(1/2,hold=True)
gamma(1/2)
sage: gamma(1/2)
sqrt(pi)

To then evaluate again, we currently must use Maxima via simplify():

sage: a = SR(1/2).gamma(hold=True); a.simplify()
sqrt(pi)

gcd(b)

Return the gcd of self and b, which must be integers or polynomials over the rational numbers.

TODO: I tried the massive gcd from http://trac.sagemath.org/sage_trac/ticket/694 on Ginac dies after about 10 seconds. Singular easily does that GCD now. Since Ginac only handles poly gcd over QQ, we should change ginac itself to use Singular.

EXAMPLES:

sage: var('x,y')
(x, y)
sage: SR(10).gcd(SR(15))
5
sage: (x^3 - 1).gcd(x-1)
x - 1
sage: (x^3 - 1).gcd(x^2+x+1)
x^2 + x + 1
sage: (x^3 - sage.symbolic.constants.pi).gcd(x-sage.symbolic.constants.pi)
Traceback (most recent call last):
...
ValueError: gcd: arguments must be polynomials over the rationals
sage: gcd(x^3 - y^3, x-y)
-x + y
sage: gcd(x^100-y^100, x^10-y^10)
-x^10 + y^10
sage: gcd(expand( (x^2+17*x+3/7*y)*(x^5 - 17*y + 2/3) ), expand((x^13+17*x+3/7*y)*(x^5 - 17*y + 2/3)) )
1/7*x^5 - 17/7*y + 2/21

gradient(variables=None)

Compute the gradient of a symbolic function.

This function returns a vector whose components are the derivatives of the original function with respect to the arguments of the original function. Alternatively, you can specify the variables as a list.

EXAMPLES:

sage: x,y = var('x y')
sage: f = x^2+y^2
sage: f.gradient()
(2*x, 2*y)
sage: g(x,y) = x^2+y^2
sage: g.gradient()
(x, y) |--> (2*x, 2*y)
sage: n = var('n')
sage: f(x,y) = x^n+y^n
sage: f.gradient()
(x, y) |--> (n*x^(n - 1), n*y^(n - 1))
sage: f.gradient([y,x])
(x, y) |--> (n*y^(n - 1), n*x^(n - 1))

has(pattern)

EXAMPLES:

sage: var('x,y,a'); w0 = SR.wild(); w1 = SR.wild()
(x, y, a)
sage: (x*sin(x + y + 2*a)).has(y)
True

Here “x+y” is not a subexpression of “x+y+2*a” (which has the subexpressions “x”, “y” and “2*a”):

sage: (x*sin(x + y + 2*a)).has(x+y)
False
sage: (x*sin(x + y + 2*a)).has(x + y + w0)
True

The following fails because “2*(x+y)” automatically gets converted to “2*x+2*y” of which “x+y” is not a subexpression:

sage: (x*sin(2*(x+y) + 2*a)).has(x+y)
False

Although x^1==x and x^0==1, neither “x” nor “1” are actually of the form “x^something”:

sage: (x+1).has(x^w0)
False

Here is another possible pitfall, where the first expression matches because the term “-x” has the form “(-1)*x” in GiNaC. To check whether a polynomial contains a linear term you should use the coeff() function instead.

sage: (4*x^2 - x + 3).has(w0*x)
True
sage: (4*x^2 + x + 3).has(w0*x)
False
sage: (4*x^2 + x + 3).has(x)
True
sage: (4*x^2 - x + 3).coefficient(x,1)
-1
sage: (4*x^2 + x + 3).coefficient(x,1)
1

has_wild()

Return True if this expression contains a wildcard.

EXAMPLES:

sage: (1 + x^2).has_wild()
False
sage: (SR.wild(0) + x^2).has_wild()
True
sage: SR.wild(0).has_wild()
True

hessian()

Compute the hessian of a function. This returns a matrix components are the 2nd partial derivatives of the original function.

EXAMPLES:

sage: x,y = var('x y')
sage: f = x^2+y^2
sage: f.hessian()
[2 0]
[0 2]
sage: g(x,y) = x^2+y^2
sage: g.hessian()
[(x, y) |--> 2 (x, y) |--> 0]
[(x, y) |--> 0 (x, y) |--> 2]

hypergeometric_simplify(algorithm='maxima')

Simplify an expression containing hypergeometric functions.

INPUT:

• algorithm – (default: 'maxima') the algorithm to use for for simplification. Implemented are 'maxima', which uses Maxima’s hgfred function, and 'sage', which uses an algorithm implemented in the hypergeometric module

ALIAS: hypergeometric_simplify() and simplify_hypergeometric() are the same

EXAMPLES:

sage: hypergeometric((5, 4), (4, 1, 2, 3),
....:                x).simplify_hypergeometric()
1/144*x^2*hypergeometric((), (3, 4), x) +...
1/3*x*hypergeometric((), (2, 3), x) + hypergeometric((), (1, 2), x)
sage: (2*hypergeometric((), (), x)).simplify_hypergeometric()
2*e^x
sage: (nest(lambda y: hypergeometric([y], [1], x), 3, 1)
....:  .simplify_hypergeometric())
laguerre(-laguerre(-e^x, x), x)
sage: (nest(lambda y: hypergeometric([y], [1], x), 3, 1)
....:  .simplify_hypergeometric(algorithm='sage'))
hypergeometric((hypergeometric((e^x,), (1,), x),), (1,), x)

imag(hold=False)

Return the imaginary part of this symbolic expression.

EXAMPLES:

sage: sqrt(-2).imag_part()
sqrt(2)

We simplify \(\ln(\exp(z))\) to \(z\). This should only be for \(-\pi<{\rm Im}(z)<=\pi\), but Maxima does not have a symbolic imaginary part function, so we cannot use assume to assume that first:

sage: z = var('z')
sage: f = log(exp(z))
sage: f
log(e^z)
sage: f.simplify()
z
sage: forget()

A more symbolic example:

sage: var('a, b')
(a, b)
sage: f = log(a + b*I)
sage: f.imag_part()
arctan2(imag_part(a) + real_part(b), -imag_part(b) + real_part(a))

Using the hold parameter it is possible to prevent automatic evaluation:

sage: I.imag_part()
1
sage: I.imag_part(hold=True)
imag_part(I)

This also works using functional notation:

sage: imag_part(I,hold=True)
imag_part(I)
sage: imag_part(I)
1

To then evaluate again, we currently must use Maxima via simplify():

sage: a = I.imag_part(hold=True); a.simplify()
1

TESTS:

sage: x = var('x')
sage: x.imag_part()
imag_part(x)
sage: SR(2+3*I).imag_part()
3
sage: SR(CC(2,3)).imag_part()
3.00000000000000
sage: SR(CDF(2,3)).imag_part()
3.0

imag_part(hold=False)

Return the imaginary part of this symbolic expression.

EXAMPLES:

sage: sqrt(-2).imag_part()
sqrt(2)

We simplify \(\ln(\exp(z))\) to \(z\). This should only be for \(-\pi<{\rm Im}(z)<=\pi\), but Maxima does not have a symbolic imaginary part function, so we cannot use assume to assume that first:

sage: z = var('z')
sage: f = log(exp(z))
sage: f
log(e^z)
sage: f.simplify()
z
sage: forget()

A more symbolic example:

sage: var('a, b')
(a, b)
sage: f = log(a + b*I)
sage: f.imag_part()
arctan2(imag_part(a) + real_part(b), -imag_part(b) + real_part(a))

Using the hold parameter it is possible to prevent automatic evaluation:

sage: I.imag_part()
1
sage: I.imag_part(hold=True)
imag_part(I)

This also works using functional notation:

sage: imag_part(I,hold=True)
imag_part(I)
sage: imag_part(I)
1

To then evaluate again, we currently must use Maxima via simplify():

sage: a = I.imag_part(hold=True); a.simplify()
1

TESTS:

sage: x = var('x')
sage: x.imag_part()
imag_part(x)
sage: SR(2+3*I).imag_part()
3
sage: SR(CC(2,3)).imag_part()
3.00000000000000
sage: SR(CDF(2,3)).imag_part()
3.0

implicit_derivative(Y, X, n=1)

Return the n’th derivative of Y with respect to X given implicitly by this expression.

INPUT:

• Y - The dependent variable of the implicit expression.
• X - The independent variable with respect to which the derivative is taken.
• n - (default : 1) the order of the derivative.

EXAMPLES:

sage: var('x, y')
(x, y)
sage: f = cos(x)*sin(y)
sage: f.implicit_derivative(y, x)
sin(x)*sin(y)/(cos(x)*cos(y))
sage: g = x*y^2
sage: g.implicit_derivative(y, x, 3)
-1/4*(y + 2*y/x)/x^2 + 1/4*(2*y^2/x - y^2/x^2)/(x*y) - 3/4*y/x^3

It is an error to not include an independent variable term in the expression:

sage: (cos(x)*sin(x)).implicit_derivative(y, x)
Traceback (most recent call last):
...
ValueError: Expression cos(x)*sin(x) contains no y terms

TESTS:

sage: var('x,y')  # check that the pynac registry is not polluted
(x, y)
sage: psr = copy(sage.symbolic.ring.pynac_symbol_registry)
sage: (x^6*y^5).implicit_derivative(y, x, 3)
-792/125*y/x^3 + 12/25*(15*x^4*y^5 + 28*x^3*y^5)/(x^6*y^4) - 36/125*(20*x^5*y^4 + 43*x^4*y^4)/(x^7*y^3)
sage: psr == sage.symbolic.ring.pynac_symbol_registry
True

integral(*args, **kwds)

Compute the integral of self. Please see sage.symbolic.integration.integral.integrate() for more details.

EXAMPLES:

sage: sin(x).integral(x,0,3)
-cos(3) + 1
sage: sin(x).integral(x)
-cos(x)

TESTS:

We check that trac ticket #12438 is resolved:

sage: f(x) = x; f
x |--> x
sage: integral(f, x)
x |--> 1/2*x^2
sage: integral(f, x, 0, 1)
1/2

sage: f(x, y) = x + y
sage: f
(x, y) |--> x + y
sage: integral(f, y, 0, 1)
x |--> x + 1/2
sage: integral(f, x, 0, 1)
y |--> y + 1/2
sage: _(3)
7/2
sage: var("z")
z
sage: integral(f, z, 0, 2)
(x, y) |--> 2*x + 2*y
sage: integral(f, z)
(x, y) |--> (x + y)*z

integrate(*args, **kwds)

Compute the integral of self. Please see sage.symbolic.integration.integral.integrate() for more details.

EXAMPLES:

sage: sin(x).integral(x,0,3)
-cos(3) + 1
sage: sin(x).integral(x)
-cos(x)

TESTS:

We check that trac ticket #12438 is resolved:

sage: f(x) = x; f
x |--> x
sage: integral(f, x)
x |--> 1/2*x^2
sage: integral(f, x, 0, 1)
1/2

sage: f(x, y) = x + y
sage: f
(x, y) |--> x + y
sage: integral(f, y, 0, 1)
x |--> x + 1/2
sage: integral(f, x, 0, 1)
y |--> y + 1/2
sage: _(3)
7/2
sage: var("z")
z
sage: integral(f, z, 0, 2)
(x, y) |--> 2*x + 2*y
sage: integral(f, z)
(x, y) |--> (x + y)*z

inverse_laplace(t, s)

Return inverse Laplace transform of self. See sage.calculus.calculus.inverse_laplace

EXAMPLES:

sage: var('w, m')
(w, m)
sage: f = (1/(w^2+10)).inverse_laplace(w, m); f
1/10*sqrt(10)*sin(sqrt(10)*m)

is_algebraic()

Return True if this expression is known to be algebraic.

EXAMPLES:

sage: sqrt(2).is_algebraic()
True
sage: (5*sqrt(2)).is_algebraic()
True
sage: (sqrt(2) + 2^(1/3) - 1).is_algebraic()
True
sage: (I*golden_ratio + sqrt(2)).is_algebraic()
True
sage: (sqrt(2) + pi).is_algebraic()
False
sage: SR(QQ(2/3)).is_algebraic()
True
sage: SR(1.2).is_algebraic()
False

is_constant()

Return True if this symbolic expression is a constant.

This function is intended to provide an interface to query the internal representation of the expression. In this sense, the word constant does not reflect the mathematical properties of the expression. Expressions which have no variables may return False.

EXAMPLES:

sage: pi.is_constant()
True
sage: x.is_constant()
False
sage: SR(1).is_constant()
False

Note that the complex I is not a constant:

sage: I.is_constant()
False
sage: I.is_numeric()
True

is_infinity()

Return True if self is an infinite expression.

EXAMPLES:

sage: SR(oo).is_infinity()
True
sage: x.is_infinity()
False

is_integer()

Return True if this expression is known to be an integer.

EXAMPLES:

sage: SR(5).is_integer()
True

is_negative()

Return True if this expression is known to be negative.

EXAMPLES:

sage: SR(-5).is_negative()
True

Check if we can correctly deduce negativity of mul objects:

sage: t0 = SR.symbol("t0", domain='positive')
sage: t0.is_negative()
False
sage: (-t0).is_negative()
True
sage: (-pi).is_negative()
True

is_negative_infinity()

Return True if self is a negative infinite expression.

EXAMPLES:

sage: SR(oo).is_negative_infinity()
False
sage: SR(-oo).is_negative_infinity()
True
sage: x.is_negative_infinity()
False

is_numeric()

A Pynac numeric is an object you can do arithmetic with that is not a symbolic variable, function, or constant. Return True if this expression only consists of a numeric object.

EXAMPLES:

sage: SR(1).is_numeric()
True
sage: x.is_numeric()
False
sage: pi.is_numeric()
False
sage: sin(x).is_numeric()
False

is_polynomial(var)

Return True if self is a polynomial in the given variable.

EXAMPLES:

sage: var('x,y,z')
(x, y, z)
sage: t = x^2 + y; t
x^2 + y
sage: t.is_polynomial(x)
True
sage: t.is_polynomial(y)
True
sage: t.is_polynomial(z)
True

sage: t = sin(x) + y; t
y + sin(x)
sage: t.is_polynomial(x)
False
sage: t.is_polynomial(y)
True
sage: t.is_polynomial(sin(x))
True

TESTS:

Check if we can handle derivatives. trac ticket #6523:

sage: f(x) = function('f',x)
sage: f(x).diff(x).is_zero()
False

Check if trac ticket #11352 is fixed:

sage: el = -1/2*(2*x^2 - sqrt(2*x - 1)*sqrt(2*x + 1) - 1)
sage: el.is_polynomial(x)
False

Check that negative exponents are handled (trac ticket #15304):

sage: y = var('y')
sage: (y/x).is_polynomial(x)
False

is_positive()

Return True if this expression is known to be positive.

EXAMPLES:

sage: t0 = SR.symbol("t0", domain='positive')
sage: t0.is_positive()
True
sage: t0.is_negative()
False
sage: t0.is_real()
True
sage: t1 = SR.symbol("t1", domain='positive')
sage: (t0*t1).is_positive()
True
sage: (t0 + t1).is_positive()
True
sage: (t0*x).is_positive()
False

is_positive_infinity()

Return True if self is a positive infinite expression.

EXAMPLES:

sage: SR(oo).is_positive_infinity()
True
sage: SR(-oo).is_positive_infinity()
False
sage: x.is_infinity()
False

is_real()

Return True if this expression is known to be a real number.

EXAMPLES:

sage: t0 = SR.symbol("t0", domain='real')
sage: t0.is_real()
True
sage: t0.is_positive()
False
sage: t1 = SR.symbol("t1", domain='positive')
sage: (t0+t1).is_real()
True
sage: (t0+x).is_real()
False
sage: (t0*t1).is_real()
True
sage: (t0*x).is_real()
False

The following is real, but we cannot deduce that.:

sage: (x*x.conjugate()).is_real()
False

is_relational()

Return True if self is a relational expression.

EXAMPLES:

sage: x = var('x')
sage: eqn = (x-1)^2 == x^2 - 2*x + 3
sage: eqn.is_relational()
True
sage: sin(x).is_relational()
False

is_series()

Return True if self is a series.

Series are special kinds of symbolic expressions that are constructed via the series() method. They usually have an Order() term unless the series representation is exact, see is_terminating_series().

OUTPUT:

Boolean. Whether self is a series symbolic expression. Usually, this means that it was constructed by the series() method.

Returns False if only a subexpression of the symbolic expression is a series.

EXAMPLES:

sage: SR(5).is_series()
False
sage: var('x')
x
sage: x.is_series()
False
sage: exp(x).is_series()
False
sage: exp(x).series(x,10).is_series()
True

Laurent series are series, too:

sage: laurent_series = (cos(x)/x).series(x, 5)
sage: laurent_series
1*x^(-1) + (-1/2)*x + 1/24*x^3 + Order(x^5)
sage: laurent_series.is_series()
True

Something only containing a series as a subexpression is not a series:

sage: sum_expr = 1 + exp(x).series(x,5); sum_expr
(1 + 1*x + 1/2*x^2 + 1/6*x^3 + 1/24*x^4 + Order(x^5)) + 1
sage: sum_expr.is_series()
False

is_symbol()

Return True if this symbolic expression consists of only a symbol, i.e., a symbolic variable.

EXAMPLES:

sage: x.is_symbol()
True
sage: var('y')
y
sage: y.is_symbol()
True
sage: (x*y).is_symbol()
False
sage: pi.is_symbol()
False

sage: ((x*y)/y).is_symbol()
True
sage: (x^y).is_symbol()
False

is_terminating_series()

Return True if self is a series without order term.

A series is terminating if it can be represented exactly, without requiring an order term. See also is_series() for general series.

OUTPUT:

Boolean. Whether self was constructed by series() and has no order term.

EXAMPLES:

sage: (x^5+x^2+1).series(x,10)
1 + 1*x^2 + 1*x^5
sage: (x^5+x^2+1).series(x,10).is_series()
True
sage: (x^5+x^2+1).series(x,10).is_terminating_series()
True
sage: SR(5).is_terminating_series()
False
sage: var('x')
x
sage: x.is_terminating_series()
False
sage: exp(x).series(x,10).is_terminating_series()
False

is_trivial_zero()

Check if this expression is trivially equal to zero without any simplification.

This method is intended to be used in library code where trying to obtain a mathematically correct result by applying potentially expensive rewrite rules is not desirable.

EXAMPLES:

sage: SR(0).is_trivial_zero()
True
sage: SR(0.0).is_trivial_zero()
True
sage: SR(float(0.0)).is_trivial_zero()
True

sage: (SR(1)/2^1000).is_trivial_zero()
False
sage: SR(1./2^10000).is_trivial_zero()
False

The is_zero() method is more capable:

sage: t = pi + (pi - 1)*pi - pi^2
sage: t.is_trivial_zero()
False
sage: t.is_zero()
True
sage: u = sin(x)^2 + cos(x)^2 - 1
sage: u.is_trivial_zero()
False
sage: u.is_zero()
True

is_unit()

Return True if this expression is a unit of the symbolic ring.

EXAMPLES:

sage: SR(1).is_unit()
True
sage: SR(-1).is_unit()
True
sage: SR(0).is_unit()
False

iterator()

Return an iterator over the operands of this expression.

EXAMPLES:

sage: x,y,z = var('x,y,z')
sage: list((x+y+z).iterator())
[x, y, z]
sage: list((x*y*z).iterator())
[x, y, z]
sage: list((x^y*z*(x+y)).iterator())
[x + y, x^y, z]

Note that symbols, constants and numeric objects do not have operands, so the iterator function raises an error in these cases:

sage: x.iterator()
Traceback (most recent call last):
...
ValueError: expressions containing only a numeric coefficient, constant or symbol have no operands
sage: pi.iterator()
Traceback (most recent call last):
...
ValueError: expressions containing only a numeric coefficient, constant or symbol have no operands
sage: SR(5).iterator()
Traceback (most recent call last):
...
ValueError: expressions containing only a numeric coefficient, constant or symbol have no operands

laplace(t, s)

Return Laplace transform of self. See sage.calculus.calculus.laplace

EXAMPLES:

sage: var('x,s,z')
(x, s, z)
sage: (z + exp(x)).laplace(x, s)
z/s + 1/(s - 1)

lcm(b)

Return the lcm of self and b, which must be integers or polynomials over the rational numbers. This is computed from the gcd of self and b implicitly from the relation self * b = gcd(self, b) * lcm(self, b).

Note

In agreement with the convention in use for integers, if self * b == 0, then gcd(self, b) == max(self, b) and lcm(self, b) == 0.

EXAMPLES:

sage: var('x,y')
(x, y)
sage: SR(10).lcm(SR(15))
30
sage: (x^3 - 1).lcm(x-1)
x^3 - 1
sage: (x^3 - 1).lcm(x^2+x+1)
x^3 - 1
sage: (x^3 - sage.symbolic.constants.pi).lcm(x-sage.symbolic.constants.pi)
Traceback (most recent call last):
...
ValueError: lcm: arguments must be polynomials over the rationals
sage: lcm(x^3 - y^3, x-y)
-x^3 + y^3
sage: lcm(x^100-y^100, x^10-y^10)
-x^100 + y^100
sage: lcm(expand( (x^2+17*x+3/7*y)*(x^5 - 17*y + 2/3) ), expand((x^13+17*x+3/7*y)*(x^5 - 17*y + 2/3)) )
 1/21*(21*x^18 - 357*x^13*y + 14*x^13 + 357*x^6 + 9*x^5*y -
         6069*x*y - 153*y^2 + 238*x + 6*y)*(21*x^7 + 357*x^6 +
                 9*x^5*y - 357*x^2*y + 14*x^2 - 6069*x*y -
                 153*y^2 + 238*x + 6*y)/(3*x^5 - 51*y + 2)

TESTS:

Verify that x * y = gcd(x,y) * lcm(x,y):

sage: x, y = var('x,y')
sage: LRs = [(SR(10), SR(15)), (x^3-1, x-1), (x^3-y^3, x-y), (x^3-1, x^2+x+1), (SR(0), x-y)]
sage: all((L.gcd(R) * L.lcm(R)) == L*R for L, R in LRs)
True

Make sure that the convention for what to do with the 0 is being respected:

sage: gcd(x, SR(0)), lcm(x, SR(0))
(x, 0)
sage: gcd(SR(0), SR(0)), lcm(SR(0), SR(0))
(0, 0)

leading_coeff(s)

Return the leading coefficient of s in self.

EXAMPLES:

sage: var('x,y,a')
(x, y, a)
sage: f = 100 + a*x + x^3*sin(x*y) + x*y + x/y + 2*sin(x*y)/x; f
x^3*sin(x*y) + a*x + x*y + x/y + 2*sin(x*y)/x + 100
sage: f.leading_coefficient(x)
sin(x*y)
sage: f.leading_coefficient(y)
x
sage: f.leading_coefficient(sin(x*y))
x^3 + 2/x

leading_coefficient(s)

Return the leading coefficient of s in self.

EXAMPLES:

sage: var('x,y,a')
(x, y, a)
sage: f = 100 + a*x + x^3*sin(x*y) + x*y + x/y + 2*sin(x*y)/x; f
x^3*sin(x*y) + a*x + x*y + x/y + 2*sin(x*y)/x + 100
sage: f.leading_coefficient(x)
sin(x*y)
sage: f.leading_coefficient(y)
x
sage: f.leading_coefficient(sin(x*y))
x^3 + 2/x

left()

If self is a relational expression, return the left hand side of the relation. Otherwise, raise a ValueError.

EXAMPLES:

sage: x = var('x')
sage: eqn = (x-1)^2 == x^2 - 2*x + 3
sage: eqn.left_hand_side()
(x - 1)^2
sage: eqn.lhs()
(x - 1)^2
sage: eqn.left()
(x - 1)^2

left_hand_side()

If self is a relational expression, return the left hand side of the relation. Otherwise, raise a ValueError.

EXAMPLES:

sage: x = var('x')
sage: eqn = (x-1)^2 == x^2 - 2*x + 3
sage: eqn.left_hand_side()
(x - 1)^2
sage: eqn.lhs()
(x - 1)^2
sage: eqn.left()
(x - 1)^2

lhs()

If self is a relational expression, return the left hand side of the relation. Otherwise, raise a ValueError.

EXAMPLES:

sage: x = var('x')
sage: eqn = (x-1)^2 == x^2 - 2*x + 3
sage: eqn.left_hand_side()
(x - 1)^2
sage: eqn.lhs()
(x - 1)^2
sage: eqn.left()
(x - 1)^2

limit(*args, **kwds)

Return a symbolic limit. See sage.calculus.calculus.limit

EXAMPLES:

sage: (sin(x)/x).limit(x=0)
1

list(x=None)

Return the coefficients of this symbolic expression as a polynomial in x.

INPUT:

• x – optional variable.

OUTPUT:

A list of expressions where the n-th element is the coefficient of x^n when self is seen as polynomial in x.

EXAMPLES:

sage: var('x, y, a')
(x, y, a)
sage: (x^5).list()
[0, 0, 0, 0, 0, 1]
sage: p = x - x^3 + 5/7*x^5
sage: p.list()
[0, 1, 0, -1, 0, 5/7]
sage: p = expand((x-a*sqrt(2))^2 + x + 1); p
-2*sqrt(2)*a*x + 2*a^2 + x^2 + x + 1
sage: p.list(a)
[x^2 + x + 1, -2*sqrt(2)*x, 2]
sage: s=(1/(1-x)).series(x,6); s
1 + 1*x + 1*x^2 + 1*x^3 + 1*x^4 + 1*x^5 + Order(x^6)
sage: s.list()
[1, 1, 1, 1, 1, 1]

log(b=None, hold=False)

Return the logarithm of self.

EXAMPLES:

sage: x, y = var('x, y')
sage: x.log()
log(x)
sage: (x^y + y^x).log()
log(x^y + y^x)
sage: SR(0).log()
-Infinity
sage: SR(-1).log()
I*pi
sage: SR(1).log()
0
sage: SR(1/2).log()
log(1/2)
sage: SR(0.5).log()
-0.693147180559945
sage: SR(0.5).log().exp()
0.500000000000000
sage: math.log(0.5)
-0.6931471805599453
sage: plot(lambda x: SR(x).log(), 0.1,10)
Graphics object consisting of 1 graphics primitive

To prevent automatic evaluation use the hold argument:

sage: I.log()
1/2*I*pi
sage: I.log(hold=True)
log(I)

To then evaluate again, we currently must use Maxima via simplify():

sage: a = I.log(hold=True); a.simplify()
1/2*I*pi

The hold parameter also works in functional notation:

sage: log(-1,hold=True)
log(-1)
sage: log(-1)
I*pi

TESTS:

sage: SR(oo).log()
+Infinity
sage: SR(-oo).log()
+Infinity
sage: SR(unsigned_infinity).log()
+Infinity

log_expand(algorithm='products')

Simplify symbolic expression, which can contain logs.

Expands logarithms of powers, logarithms of products and logarithms of quotients. The option algorithm specifies which expression types should be expanded.

INPUT:

• self - expression to be simplified

• algorithm - (default: ‘products’) optional, governs which expression is expanded. Possible values are

  • ‘nothing’ (no expansion),
  • ‘powers’ (log(a^r) is expanded),
  • ‘products’ (like ‘powers’ and also log(a*b) are expanded),
  • ‘all’ (all possible expansion).

  See also examples below.

DETAILS: This uses the Maxima simplifier and sets logexpand option for this simplifier. From the Maxima documentation: “Logexpand:true causes log(a^b) to become b*log(a). If it is set to all, log(a*b) will also simplify to log(a)+log(b). If it is set to super, then log(a/b) will also simplify to log(a)-log(b) for rational numbers a/b, a#1. (log(1/b), for integer b, always simplifies.) If it is set to false, all of these simplifications will be turned off. “

ALIAS: log_expand() and expand_log() are the same

EXAMPLES:

By default powers and products (and quotients) are expanded, but not quotients of integers:

sage: (log(3/4*x^pi)).log_expand()
pi*log(x) + log(3/4)

To expand also log(3/4) use algorithm='all':

sage: (log(3/4*x^pi)).log_expand('all')
pi*log(x) - log(4) + log(3)

To expand only the power use algorithm='powers'.:

sage: (log(x^6)).log_expand('powers')
6*log(x)

The expression log((3*x)^6) is not expanded with algorithm='powers', since it is converted into product first:

sage: (log((3*x)^6)).log_expand('powers')
log(729*x^6)

This shows that the option algorithm from the previous call has no influence to future calls (we changed some default Maxima flag, and have to ensure that this flag has been restored):

sage: (log(3/4*x^pi)).log_expand()
pi*log(x) + log(3/4)

sage: (log(3/4*x^pi)).log_expand('all')
pi*log(x) - log(4) + log(3)

sage: (log(3/4*x^pi)).log_expand()
pi*log(x) + log(3/4)

TESTS:

Most of these log expansions only make sense over the reals. So, we should set the Maxima domain variable to ‘real’ before we call out to Maxima. When we return, however, we should set the domain back to what it was, rather than assuming that it was ‘complex’. See trac ticket #12780:

sage: from sage.calculus.calculus import maxima
sage: maxima('domain: real;')
real
sage: x.expand_log()
x
sage: maxima('domain;')
real
sage: maxima('domain: complex;')
complex

AUTHORS:

• Robert Marik (11-2009)

log_gamma(hold=False)

Return the log gamma function evaluated at self. This is the logarithm of gamma of self, where gamma is a complex function such that \(gamma(n)\) equals \(factorial(n-1)\).

EXAMPLES:

sage: x = var('x')
sage: x.log_gamma()
log_gamma(x)
sage: SR(2).log_gamma()
0
sage: SR(5).log_gamma()
log(24)
sage: a = SR(5).log_gamma(); a.n()
3.17805383034795
sage: SR(5-1).factorial().log()
log(24)
sage: set_verbose(-1); plot(lambda x: SR(x).log_gamma(), -7,8, plot_points=1000).show()
sage: math.exp(0.5)
1.6487212707001282
sage: plot(lambda x: (SR(x).exp() - SR(-x).exp())/2 - SR(x).sinh(), -1, 1)
Graphics object consisting of 1 graphics primitive

To prevent automatic evaluation use the hold argument:

sage: SR(5).log_gamma(hold=True)
log_gamma(5)

To evaluate again, currently we must use numerical evaluation via n():

sage: a = SR(5).log_gamma(hold=True); a.n()
3.17805383034795

log_simplify(algorithm=None)

Simplify a (real) symbolic expression that contains logarithms.

The given expression is scanned recursively, transforming subexpressions of the form \(a \log(b) + c \log(d)\) into \(\log(b^{a} d^{c})\) before simplifying within the log().

The user can specify conditions that \(a\) and \(c\) must satisfy before this transformation will be performed using the optional parameter algorithm.

Warning

This is only safe to call if every variable in the given expression is assumed to be real. The simplification it performs is in general not valid over the complex numbers. For example:

sage: x,y = SR.var('x,y')
sage: f = log(x*y) - (log(x) + log(y))
sage: f(x=-1, y=i)
-2*I*pi
sage: f.simplify_log()
0

INPUT:

• self - expression to be simplified
• algorithm - (default: None) optional, governs the condition on \(a\) and \(c\) which must be satisfied to contract expression \(a \log(b) + c \log(d)\). Values are
  • None (use Maxima default, integers),
  • 'one' (1 and -1),
  • 'ratios' (rational numbers),
  • 'constants' (constants),
  • 'all' (all expressions).

ALGORITHM:

This uses the Maxima logcontract() command.

ALIAS:

log_simplify() and simplify_log() are the same.

EXAMPLES:

sage: x,y,t=var('x y t')

Only two first terms are contracted in the following example; the logarithm with coefficient \(\frac{1}{2}\) is not contracted:

sage: f = log(x)+2*log(y)+1/2*log(t)
sage: f.simplify_log()
log(x*y^2) + 1/2*log(t)

To contract all terms in the previous example, we use the 'ratios' algorithm:

sage: f.simplify_log(algorithm='ratios')
log(sqrt(t)*x*y^2)

To contract terms with no coefficient (more precisely, with coefficients \(1\) and \(-1\)), we use the 'one' algorithm:

sage: f = log(x)+2*log(y)-log(t)
sage: f.simplify_log('one')
2*log(y) + log(x/t)

sage: f = log(x)+log(y)-1/3*log((x+1))
sage: f.simplify_log()
log(x*y) - 1/3*log(x + 1)

sage: f.simplify_log('ratios')
log(x*y/(x + 1)^(1/3))

\(\pi\) is an irrational number; to contract logarithms in the following example we have to set algorithm to 'constants' or 'all':

sage: f = log(x)+log(y)-pi*log((x+1))
sage: f.simplify_log('constants')
log(x*y/(x + 1)^pi)

x*log(9) is contracted only if algorithm is 'all':

sage: (x*log(9)).simplify_log()
x*log(9)
sage: (x*log(9)).simplify_log('all')
log(9^x)

TESTS:

Ensure that the option algorithm from one call has no influence upon future calls (a Maxima flag was set, and we have to ensure that its value has been restored):

sage: f = log(x)+2*log(y)+1/2*log(t)
sage: f.simplify_log('one')
1/2*log(t) + log(x) + 2*log(y)

sage: f.simplify_log('ratios')
log(sqrt(t)*x*y^2)

sage: f.simplify_log()
log(x*y^2) + 1/2*log(t)

This shows that the issue at trac ticket #7334 is fixed. Maxima intentionally keeps the expression inside the log factored:

sage: log_expr = (log(sqrt(2)-1)+log(sqrt(2)+1))
sage: log_expr.simplify_log('all')
log((sqrt(2) + 1)*(sqrt(2) - 1))
sage: _.simplify_rational()
0

We should use the current simplification domain rather than set it to ‘real’ explicitly (trac ticket #12780):

sage: f = sqrt(x^2)
sage: f.simplify_log()
sqrt(x^2)
sage: from sage.calculus.calculus import maxima
sage: maxima('domain: real;')
real
sage: f.simplify_log()
abs(x)
sage: maxima('domain: complex;')
complex

AUTHORS:

• Robert Marik (11-2009)

low_degree(s)

Return the exponent of the lowest nonpositive power of s in self.

OUTPUT:

An integer <= 0.

EXAMPLES:

sage: var('x,y,a')
(x, y, a)
sage: f = 100 + a*x + x^3*sin(x*y) + x*y + x/y^10 + 2*sin(x*y)/x; f
x^3*sin(x*y) + a*x + x*y + 2*sin(x*y)/x + x/y^10 + 100
sage: f.low_degree(x)
-1
sage: f.low_degree(y)
-10
sage: f.low_degree(sin(x*y))
0
sage: (x^3+y).low_degree(x)
0

match(pattern)

Check if self matches the given pattern.

INPUT:

• pattern – a symbolic expression, possibly containing wildcards to match for

OUTPUT:

One of

None if there is no match, or a dictionary mapping the wildcards to the matching values if a match was found. Note that the dictionary is empty if there were no wildcards in the given pattern.

See also http://www.ginac.de/tutorial/Pattern-matching-and-advanced-substitutions.html

EXAMPLES:

sage: var('x,y,z,a,b,c,d,f,g')
(x, y, z, a, b, c, d, f, g)
sage: w0 = SR.wild(0); w1 = SR.wild(1); w2 = SR.wild(2)
sage: ((x+y)^a).match((x+y)^a)  # no wildcards, so empty dict
{}
sage: print ((x+y)^a).match((x+y)^b)
None
sage: t = ((x+y)^a).match(w0^w1)
sage: t[w0], t[w1]
(x + y, a)
sage: print ((x+y)^a).match(w0^w0)
None
sage: ((x+y)^(x+y)).match(w0^w0)
{$0: x + y}
sage: t = ((a+b)*(a+c)).match((a+w0)*(a+w1))
sage: t[w0], t[w1]
(c, b)
sage: ((a+b)*(a+c)).match((w0+b)*(w0+c))
{$0: a}
sage: t = ((a+b)*(a+c)).match((w0+w1)*(w0+w2))
sage: t[w0], t[w1], t[w2]
(a, c, b)
sage: print ((a+b)*(a+c)).match((w0+w1)*(w1+w2))
None
sage: t = (a*(x+y)+a*z+b).match(a*w0+w1)
sage: t[w0], t[w1]
(x + y, a*z + b)
sage: print (a+b+c+d+f+g).match(c)
None
sage: (a+b+c+d+f+g).has(c)
True
sage: (a+b+c+d+f+g).match(c+w0)
{$0: a + b + d + f + g}
sage: (a+b+c+d+f+g).match(c+g+w0)
{$0: a + b + d + f}
sage: (a+b).match(a+b+w0)
{$0: 0}
sage: print (a*b^2).match(a^w0*b^w1)
None
sage: (a*b^2).match(a*b^w1)
{$1: 2}
sage: (x*x.arctan2(x^2)).match(w0*w0.arctan2(w0^2))
{$0: x}

Beware that behind-the-scenes simplification can lead to surprising results in matching:

sage: print (x+x).match(w0+w1)
None
sage: t = x+x; t
2*x
sage: t.operator()
<function mul_vararg ...>

Since asking to match w0+w1 looks for an addition operator, there is no match.

maxima_methods()

Provide easy access to maxima methods, converting the result to a Sage expression automatically.

EXAMPLES:

sage: t = log(sqrt(2) - 1) + log(sqrt(2) + 1); t
log(sqrt(2) + 1) + log(sqrt(2) - 1)
sage: res = t.maxima_methods().logcontract(); res
log((sqrt(2) + 1)*(sqrt(2) - 1))
sage: type(res)
<type 'sage.symbolic.expression.Expression'>

minpoly(*args, **kwds)

Return the minimal polynomial of this symbolic expression.

EXAMPLES:

sage: golden_ratio.minpoly()
x^2 - x - 1

mul(hold=False, *args)

Return the product of the current expression and the given arguments.

To prevent automatic evaluation use the hold argument.

EXAMPLES:

sage: x.mul(x)
x^2
sage: x.mul(x, hold=True)
x*x
sage: x.mul(x, (2+x), hold=True)
(x + 2)*x*x
sage: x.mul(x, (2+x), x, hold=True)
(x + 2)*x*x*x
sage: x.mul(x, (2+x), x, 2*x, hold=True)
(2*x)*(x + 2)*x*x*x

To then evaluate again, we currently must use Maxima via simplify():

sage: a = x.mul(x, hold=True); a.simplify()
x^2

multiply_both_sides(x, checksign=None)

Return a relation obtained by multiplying both sides of this relation by x.

Note

The checksign keyword argument is currently ignored and is included for backward compatibility reasons only.

EXAMPLES:

sage: var('x,y'); f = x + 3 < y - 2
(x, y)
sage: f.multiply_both_sides(7)
7*x + 21 < 7*y - 14
sage: f.multiply_both_sides(-1/2)
-1/2*x - 3/2 < -1/2*y + 1
sage: f*(-2/3)
-2/3*x - 2 < -2/3*y + 4/3
sage: f*(-pi)
-pi*(x + 3) < -pi*(y - 2)

Since the direction of the inequality never changes when doing arithmetic with equations, you can multiply or divide the equation by a quantity with unknown sign:

sage: f*(1+I)
(I + 1)*x + 3*I + 3 < (I + 1)*y - 2*I - 2
sage: f = sqrt(2) + x == y^3
sage: f.multiply_both_sides(I)
I*x + I*sqrt(2) == I*y^3
sage: f.multiply_both_sides(-1)
-x - sqrt(2) == -y^3

Note that the direction of the following inequalities is not reversed:

sage: (x^3 + 1 > 2*sqrt(3)) * (-1)
-x^3 - 1 > -2*sqrt(3)
sage: (x^3 + 1 >= 2*sqrt(3)) * (-1)
-x^3 - 1 >= -2*sqrt(3)
sage: (x^3 + 1 <= 2*sqrt(3)) * (-1)
-x^3 - 1 <= -2*sqrt(3)

n(prec=None, digits=None, algorithm=None)

Return a numerical approximation this symbolic expression as either a real or complex number with at least the requested number of bits or digits of precision.

EXAMPLES:

sage: sin(x).subs(x=5).n()
-0.958924274663138
sage: sin(x).subs(x=5).n(100)
-0.95892427466313846889315440616
sage: sin(x).subs(x=5).n(digits=50)
-0.95892427466313846889315440615599397335246154396460
sage: zeta(x).subs(x=2).numerical_approx(digits=50)
1.6449340668482264364724151666460251892189499012068

sage: cos(3).numerical_approx(200)
-0.98999249660044545727157279473126130239367909661558832881409
sage: numerical_approx(cos(3),200)
-0.98999249660044545727157279473126130239367909661558832881409
sage: numerical_approx(cos(3), digits=10)
-0.9899924966
sage: (i + 1).numerical_approx(32)
1.00000000 + 1.00000000*I
sage: (pi + e + sqrt(2)).numerical_approx(100)
7.2740880444219335226246195788

TESTS:

We test the evaluation of different infinities available in Pynac:

sage: t = x - oo; t
-Infinity
sage: t.n()
-infinity
sage: t = x + oo; t
+Infinity
sage: t.n()
+infinity
sage: t = x - unsigned_infinity; t
Infinity
sage: t.n()
Traceback (most recent call last):
...
ValueError: can only convert signed infinity to RR

Some expressions cannot be evaluated numerically:

sage: n(sin(x))
Traceback (most recent call last):
...
TypeError: cannot evaluate symbolic expression numerically
sage: a = var('a')
sage: (x^2 + 2*x + 2).subs(x=a).n()
Traceback (most recent call last):
...
TypeError: cannot evaluate symbolic expression numerically

Make sure we’ve rounded up log(10,2) enough to guarantee sufficient precision (trac ticket #10164):

sage: ks = 4*10**5, 10**6
sage: all(len(str(e.n(digits=k)))-1 >= k for k in ks)
True

negation()

Return the negated version of self, that is the relation that is False iff self is True.

EXAMPLES:

sage: (x < 5).negation()
x >= 5
sage: (x == sin(3)).negation()
x != sin(3)
sage: (2*x >= sqrt(2)).negation()
2*x < sqrt(2)

nintegral(*args, **kwds)

Compute the numerical integral of self. Please see sage.calculus.calculus.nintegral for more details.

EXAMPLES:

sage: sin(x).nintegral(x,0,3)
(1.989992496600..., 2.209335488557...e-14, 21, 0)

nintegrate(*args, **kwds)

Compute the numerical integral of self. Please see sage.calculus.calculus.nintegral for more details.

EXAMPLES:

sage: sin(x).nintegral(x,0,3)
(1.989992496600..., 2.209335488557...e-14, 21, 0)

nops()

Return the number of arguments of this expression.

EXAMPLES:

sage: var('a,b,c,x,y')
(a, b, c, x, y)
sage: a.number_of_operands()
0
sage: (a^2 + b^2 + (x+y)^2).number_of_operands()
3
sage: (a^2).number_of_operands()
2
sage: (a*b^2*c).number_of_operands()
3

norm()

Return the complex norm of this symbolic expression, i.e., the expression times its complex conjugate. If \(c = a + bi\) is a complex number, then the norm of \(c\) is defined as the product of \(c\) and its complex conjugate

\[\text{norm}(c) = \text{norm}(a + bi) = c \cdot \overline{c} = a^2 + b^2.\]

The norm of a complex number is different from its absolute value. The absolute value of a complex number is defined to be the square root of its norm. A typical use of the complex norm is in the integral domain \(\ZZ[i]\) of Gaussian integers, where the norm of each Gaussian integer \(c = a + bi\) is defined as its complex norm.

See also

• sage.misc.functional.norm()

EXAMPLES:

sage: a = 1 + 2*I
sage: a.norm()
5
sage: a = sqrt(2) + 3^(1/3)*I; a
sqrt(2) + I*3^(1/3)
sage: a.norm()
3^(2/3) + 2
sage: CDF(a).norm()
4.080083823051...
sage: CDF(a.norm())
4.080083823051904

normalize()

Return this expression normalized as a fraction

EXAMPLES:

sage: var('x, y, a, b, c')
(x, y, a, b, c)
sage: g = x + y/(x + 2)
sage: g.normalize()
(x^2 + 2*x + y)/(x + 2)

sage: f = x*(x-1)/(x^2 - 7) + y^2/(x^2-7) + 1/(x+1) + b/a + c/a
sage: f.normalize()
(a*x^3 + b*x^3 + c*x^3 + a*x*y^2 + a*x^2 + b*x^2 + c*x^2 +
        a*y^2 - a*x - 7*b*x - 7*c*x - 7*a - 7*b - 7*c)/((x^2 -
            7)*a*(x + 1))

ALGORITHM: Uses GiNaC.

number_of_arguments()

EXAMPLES:

sage: x,y = var('x,y')
sage: f = x + y
sage: f.number_of_arguments()
2

sage: g = f.function(x)
sage: g.number_of_arguments()
1

sage: x,y,z = var('x,y,z')
sage: (x+y).number_of_arguments()
2
sage: (x+1).number_of_arguments()
1
sage: (sin(x)+1).number_of_arguments()
1
sage: (sin(z)+x+y).number_of_arguments()
3
sage: (sin(x+y)).number_of_arguments()
2

sage: ( 2^(8/9) - 2^(1/9) )(x-1)
Traceback (most recent call last):
...
ValueError: the number of arguments must be less than or equal to 0

number_of_operands()

Return the number of arguments of this expression.

EXAMPLES:

sage: var('a,b,c,x,y')
(a, b, c, x, y)
sage: a.number_of_operands()
0
sage: (a^2 + b^2 + (x+y)^2).number_of_operands()
3
sage: (a^2).number_of_operands()
2
sage: (a*b^2*c).number_of_operands()
3

numerator(normalize=True)

Return the numerator of this symbolic expression

INPUT:

• normalize – (default: True) a boolean.

If normalize is True, the expression is first normalized to have it as a fraction before getting the numerator.

If normalize is False, the expression is kept and if it is not a quotient, then this will return the expression itself.

See also

normalize(), denominator(), numerator_denominator(), combine()

EXAMPLES:

sage: a, x, y = var('a,x,y')
sage: f = x*(x-a)/((x^2 - y)*(x-a)); f
x/(x^2 - y)
sage: f.numerator()
x
sage: f.denominator()
x^2 - y
sage: f.numerator(normalize=False)
x
sage: f.denominator(normalize=False)
x^2 - y

sage: y = var('y')
sage: g = x + y/(x + 2); g
x + y/(x + 2)
sage: g.numerator()
x^2 + 2*x + y
sage: g.denominator()
x + 2
sage: g.numerator(normalize=False)
x + y/(x + 2)
sage: g.denominator(normalize=False)
1

TESTS:

sage: ((x+y)^2/(x-y)^3*x^3).numerator(normalize=False)
(x + y)^2*x^3
sage: ((x+y)^2*x^3).numerator(normalize=False)
(x + y)^2*x^3
sage: (y/x^3).numerator(normalize=False)
y
sage: t = y/x^3/(x+y)^(1/2); t
y/(sqrt(x + y)*x^3)
sage: t.numerator(normalize=False)
y
sage: (1/x^3).numerator(normalize=False)
1
sage: (x^3).numerator(normalize=False)
x^3
sage: (y*x^sin(x)).numerator(normalize=False)
Traceback (most recent call last):
...
TypeError: self is not a rational expression

numerator_denominator(normalize=True)

Return the numerator and the denominator of this symbolic expression

INPUT:

• normalize – (default: True) a boolean.

If normalize is True, the expression is first normalized to have it as a fraction before getting the numerator and denominator.

If normalize is False, the expression is kept and if it is not a quotient, then this will return the expression itself together with 1.

See also

normalize(), numerator(), denominator(), combine()

EXAMPLE:

sage: x, y, a = var("x y a")
sage: ((x+y)^2/(x-y)^3*x^3).numerator_denominator()
((x + y)^2*x^3, (x - y)^3)

sage: ((x+y)^2/(x-y)^3*x^3).numerator_denominator(False)
((x + y)^2*x^3, (x - y)^3)

sage: g = x + y/(x + 2)
sage: g.numerator_denominator()
(x^2 + 2*x + y, x + 2)
sage: g.numerator_denominator(normalize=False)
(x + y/(x + 2), 1)

sage: g = x^2*(x + 2)
sage: g.numerator_denominator()
((x + 2)*x^2, 1)
sage: g.numerator_denominator(normalize=False)
((x + 2)*x^2, 1)

TESTS:

sage: ((x+y)^2/(x-y)^3*x^3).numerator_denominator(normalize=False)
((x + y)^2*x^3, (x - y)^3)
sage: ((x+y)^2*x^3).numerator_denominator(normalize=False)
((x + y)^2*x^3, 1)
sage: (y/x^3).numerator_denominator(normalize=False)
(y, x^3)
sage: t = y/x^3/(x+y)^(1/2); t
y/(sqrt(x + y)*x^3)
sage: t.numerator_denominator(normalize=False)
(y, sqrt(x + y)*x^3)
sage: (1/x^3).numerator_denominator(normalize=False)
(1, x^3)
sage: (x^3).numerator_denominator(normalize=False)
(x^3, 1)
sage: (y*x^sin(x)).numerator_denominator(normalize=False)
Traceback (most recent call last):
...
TypeError: self is not a rational expression

numerical_approx(prec=None, digits=None, algorithm=None)

Return a numerical approximation this symbolic expression as either a real or complex number with at least the requested number of bits or digits of precision.

EXAMPLES:

sage: sin(x).subs(x=5).n()
-0.958924274663138
sage: sin(x).subs(x=5).n(100)
-0.95892427466313846889315440616
sage: sin(x).subs(x=5).n(digits=50)
-0.95892427466313846889315440615599397335246154396460
sage: zeta(x).subs(x=2).numerical_approx(digits=50)
1.6449340668482264364724151666460251892189499012068

sage: cos(3).numerical_approx(200)
-0.98999249660044545727157279473126130239367909661558832881409
sage: numerical_approx(cos(3),200)
-0.98999249660044545727157279473126130239367909661558832881409
sage: numerical_approx(cos(3), digits=10)
-0.9899924966
sage: (i + 1).numerical_approx(32)
1.00000000 + 1.00000000*I
sage: (pi + e + sqrt(2)).numerical_approx(100)
7.2740880444219335226246195788

TESTS:

We test the evaluation of different infinities available in Pynac:

sage: t = x - oo; t
-Infinity
sage: t.n()
-infinity
sage: t = x + oo; t
+Infinity
sage: t.n()
+infinity
sage: t = x - unsigned_infinity; t
Infinity
sage: t.n()
Traceback (most recent call last):
...
ValueError: can only convert signed infinity to RR

Some expressions cannot be evaluated numerically:

sage: n(sin(x))
Traceback (most recent call last):
...
TypeError: cannot evaluate symbolic expression numerically
sage: a = var('a')
sage: (x^2 + 2*x + 2).subs(x=a).n()
Traceback (most recent call last):
...
TypeError: cannot evaluate symbolic expression numerically

Make sure we’ve rounded up log(10,2) enough to guarantee sufficient precision (trac ticket #10164):

sage: ks = 4*10**5, 10**6
sage: all(len(str(e.n(digits=k)))-1 >= k for k in ks)
True

operands()

Return a list containing the operands of this expression.

EXAMPLES:

sage: var('a,b,c,x,y')
(a, b, c, x, y)
sage: (a^2 + b^2 + (x+y)^2).operands()
[a^2, b^2, (x + y)^2]
sage: (a^2).operands()
[a, 2]
sage: (a*b^2*c).operands()
[a, b^2, c]

operator()

Return the topmost operator in this expression.

EXAMPLES:

sage: x,y,z = var('x,y,z')
sage: (x+y).operator()
<function add_vararg ...>
sage: (x^y).operator()
<built-in function pow>
sage: (x^y * z).operator()
<function mul_vararg ...>
sage: (x < y).operator()
<built-in function lt>

sage: abs(x).operator()
abs
sage: r = gamma(x).operator(); type(r)
<class 'sage.functions.other.Function_gamma'>

sage: psi = function('psi', nargs=1)
sage: psi(x).operator()
psi

sage: r = psi(x).operator()
sage: r == psi
True

sage: f = function('f', nargs=1, conjugate_func=lambda self, x: 2*x)
sage: nf = f(x).operator()
sage: nf(x).conjugate()
2*x

sage: f = function('f')
sage: a = f(x).diff(x); a
D[0](f)(x)
sage: a.operator()
D[0](f)

TESTS:

sage: (x <= y).operator()
<built-in function le>
sage: (x == y).operator()
<built-in function eq>
sage: (x != y).operator()
<built-in function ne>
sage: (x > y).operator()
<built-in function gt>
sage: (x >= y).operator()
<built-in function ge>
sage: SR._force_pyobject( (x, x + 1, x + 2) ).operator()
<type 'tuple'>

partial_fraction(var=None)

Return the partial fraction expansion of self with respect to the given variable.

INPUT:

• var - variable name or string (default: first variable)

OUTPUT:

A symbolic expression.

EXAMPLES:

sage: f = x^2/(x+1)^3
sage: f.partial_fraction()
1/(x + 1) - 2/(x + 1)^2 + 1/(x + 1)^3
sage: f.partial_fraction()
1/(x + 1) - 2/(x + 1)^2 + 1/(x + 1)^3

Notice that the first variable in the expression is used by default:

sage: y = var('y')
sage: f = y^2/(y+1)^3
sage: f.partial_fraction()
1/(y + 1) - 2/(y + 1)^2 + 1/(y + 1)^3

sage: f = y^2/(y+1)^3 + x/(x-1)^3
sage: f.partial_fraction()
y^2/(y^3 + 3*y^2 + 3*y + 1) + 1/(x - 1)^2 + 1/(x - 1)^3

You can explicitly specify which variable is used:

sage: f.partial_fraction(y)
x/(x^3 - 3*x^2 + 3*x - 1) + 1/(y + 1) - 2/(y + 1)^2 + 1/(y + 1)^3

plot(*args, **kwds)

Plot a symbolic expression. All arguments are passed onto the standard plot command.

EXAMPLES:

This displays a straight line:

sage: sin(2).plot((x,0,3))
Graphics object consisting of 1 graphics primitive

This draws a red oscillatory curve:

sage: sin(x^2).plot((x,0,2*pi), rgbcolor=(1,0,0))
Graphics object consisting of 1 graphics primitive

Another plot using the variable theta:

sage: var('theta')
theta
sage: (cos(theta) - erf(theta)).plot((theta,-2*pi,2*pi))
Graphics object consisting of 1 graphics primitive

A very thick green plot with a frame:

sage: sin(x).plot((x,-4*pi, 4*pi), thickness=20, rgbcolor=(0,0.7,0)).show(frame=True)

You can embed 2d plots in 3d space as follows:

sage: plot(sin(x^2), (x,-pi, pi), thickness=2).plot3d(z = 1)
Graphics3d Object

A more complicated family:

sage: G = sum([plot(sin(n*x), (x,-2*pi, 2*pi)).plot3d(z=n) for n in [0,0.1,..1]])
sage: G.show(frame_aspect_ratio=[1,1,1/2])  # long time (5s on sage.math, 2012)

A plot involving the floor function:

sage: plot(1.0 - x * floor(1/x), (x,0.00001,1.0))
Graphics object consisting of 1 graphics primitive

Sage used to allow symbolic functions with “no arguments”; this no longer works:

sage: plot(2*sin, -4, 4)
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for '*': 'Integer Ring' and '<class 'sage.functions.trig.Function_sin'>'

You should evaluate the function first:

sage: plot(2*sin(x), -4, 4)
Graphics object consisting of 1 graphics primitive

TESTS:

sage: f(x) = x*(1 - x)
sage: plot(f,0,1)
Graphics object consisting of 1 graphics primitive

poly(x=None)

Express this symbolic expression as a polynomial in x. If this is not a polynomial in x, then some coefficients may be functions of x.

Warning

This is different from polynomial() which returns a Sage polynomial over a given base ring.

EXAMPLES:

sage: var('a, x')
(a, x)
sage: p = expand((x-a*sqrt(2))^2 + x + 1); p
-2*sqrt(2)*a*x + 2*a^2 + x^2 + x + 1
sage: p.poly(a)
-2*sqrt(2)*a*x + 2*a^2 + x^2 + x + 1
sage: bool(p.poly(a) == (x-a*sqrt(2))^2 + x + 1)
True
sage: p.poly(x)
2*a^2 - (2*sqrt(2)*a - 1)*x + x^2 + 1

polynomial(base_ring=None, ring=None)

Return this symbolic expression as an algebraic polynomial over the given base ring, if possible.

The point of this function is that it converts purely symbolic polynomials into optimised algebraic polynomials over a given base ring.

You can specify either the base ring (base_ring) you want the output polynomial to be over, or you can specify the full polynomial ring (ring) you want the output polynomial to be an element of.

INPUT:

• base_ring - (optional) the base ring for the polynomial
• ring - (optional) the parent for the polynomial

Warning

This is different from poly() which is used to rewrite self as a polynomial in terms of one of the variables.

EXAMPLES:

sage: f = x^2 -2/3*x + 1
sage: f.polynomial(QQ)
x^2 - 2/3*x + 1
sage: f.polynomial(GF(19))
x^2 + 12*x + 1

Polynomials can be useful for getting the coefficients of an expression:

sage: g = 6*x^2 - 5
sage: g.coefficients()
[[-5, 0], [6, 2]]
sage: g.polynomial(QQ).list()
[-5, 0, 6]
sage: g.polynomial(QQ).dict()
{0: -5, 2: 6}

sage: f = x^2*e + x + pi/e
sage: f.polynomial(RDF)  # abs tol 5e-16
2.718281828459045*x^2 + x + 1.1557273497909217
sage: g = f.polynomial(RR); g
2.71828182845905*x^2 + x + 1.15572734979092
sage: g.parent()
Univariate Polynomial Ring in x over Real Field with 53 bits of precision
sage: f.polynomial(RealField(100))
2.7182818284590452353602874714*x^2 + x + 1.1557273497909217179100931833
sage: f.polynomial(CDF)  # abs tol 5e-16
2.718281828459045*x^2 + x + 1.1557273497909217
sage: f.polynomial(CC)
2.71828182845905*x^2 + x + 1.15572734979092

We coerce a multivariate polynomial with complex symbolic coefficients:

sage: x, y, n = var('x, y, n')
sage: f = pi^3*x - y^2*e - I; f
pi^3*x - y^2*e - I
sage: f.polynomial(CDF)
(-2.71828182846)*y^2 + 31.0062766803*x - 1.0*I
sage: f.polynomial(CC)
(-2.71828182845905)*y^2 + 31.0062766802998*x - 1.00000000000000*I
sage: f.polynomial(ComplexField(70))
(-2.7182818284590452354)*y^2 + 31.006276680299820175*x - 1.0000000000000000000*I

Another polynomial:

sage: f = sum((e*I)^n*x^n for n in range(5)); f
x^4*e^4 - I*x^3*e^3 - x^2*e^2 + I*x*e + 1
sage: f.polynomial(CDF)   # abs tol 5e-16
54.598150033144236*x^4 - 20.085536923187668*I*x^3 - 7.38905609893065*x^2 + 2.718281828459045*I*x + 1.0
sage: f.polynomial(CC)
54.5981500331442*x^4 - 20.0855369231877*I*x^3 - 7.38905609893065*x^2 + 2.71828182845905*I*x + 1.00000000000000

A multivariate polynomial over a finite field:

sage: f = (3*x^5 - 5*y^5)^7; f
(3*x^5 - 5*y^5)^7
sage: g = f.polynomial(GF(7)); g
3*x^35 + 2*y^35
sage: parent(g)
Multivariate Polynomial Ring in x, y over Finite Field of size 7

We check to make sure constants are converted appropriately:

sage: (pi*x).polynomial(SR)
pi*x

Using the ring parameter, you can also create polynomials rings over the symbolic ring where only certain variables are considered generators of the polynomial ring and the others are considered “constants”:

sage: a, x, y = var('a,x,y')
sage: f = a*x^10*y+3*x
sage: B = f.polynomial(ring=SR['x,y'])
sage: B.coefficients()
[a, 3]

power(exp, hold=False)

Return the current expression to the power exp.

To prevent automatic evaluation use the hold argument.

EXAMPLES:

sage: (x^2).power(2)
x^4
sage: (x^2).power(2, hold=True)
(x^2)^2

To then evaluate again, we currently must use Maxima via simplify():

sage: a = (x^2).power(2, hold=True); a.simplify()
x^4

power_series(base_ring)

Return algebraic power series associated to this symbolic expression, which must be a polynomial in one variable, with coefficients coercible to the base ring.

The power series is truncated one more than the degree.

EXAMPLES:

sage: theta = var('theta')
sage: f = theta^3 + (1/3)*theta - 17/3
sage: g = f.power_series(QQ); g
-17/3 + 1/3*theta + theta^3 + O(theta^4)
sage: g^3
-4913/27 + 289/9*theta - 17/9*theta^2 + 2602/27*theta^3 + O(theta^4)
sage: g.parent()
Power Series Ring in theta over Rational Field

primitive_part(s)

Return the primitive polynomial of this expression when considered as a polynomial in s.

See also unit(), content(), and unit_content_primitive().

INPUT:

• s – a symbolic expression.

OUTPUT:

The primitive polynomial as a symbolic expression. It is defined as the quotient by the unit() and content() parts (with respect to the variable s).

EXAMPLES:

sage: (2*x+4).primitive_part(x)
x + 2
sage: (2*x+1).primitive_part(x)
2*x + 1
sage: (2*x+1/2).primitive_part(x)
4*x + 1
sage: var('y')
y
sage: (2*x + 4*sin(y)).primitive_part(sin(y))
x + 2*sin(y)

pyobject()

Get the underlying Python object.

OUTPUT:

The Python object corresponding to this expression, assuming this expression is a single numerical value or an infinity representable in Python. Otherwise, a TypeError is raised.

EXAMPLES:

sage: var('x')
x
sage: b = -17/3
sage: a = SR(b)
sage: a.pyobject()
-17/3
sage: a.pyobject() is b
True

TESTS:

sage: SR(oo).pyobject()
+Infinity
sage: SR(-oo).pyobject()
-Infinity
sage: SR(unsigned_infinity).pyobject()
Infinity
sage: SR(I*oo).pyobject()
Traceback (most recent call last):
...
TypeError: Python infinity cannot have complex phase.

radical_simplify(*args, **kwds)

Deprecated: Use canonicalize_radical() instead. See trac ticket #11912 for details.

rational_expand(side=None)

Expand this symbolic expression. Products of sums and exponentiated sums are multiplied out, numerators of rational expressions which are sums are split into their respective terms, and multiplications are distributed over addition at all levels.

EXAMPLES:

We expand the expression \((x-y)^5\) using both method and functional notation.

sage: x,y = var('x,y')
sage: a = (x-y)^5
sage: a.expand()
x^5 - 5*x^4*y + 10*x^3*y^2 - 10*x^2*y^3 + 5*x*y^4 - y^5
sage: expand(a)
x^5 - 5*x^4*y + 10*x^3*y^2 - 10*x^2*y^3 + 5*x*y^4 - y^5

We expand some other expressions:

sage: expand((x-1)^3/(y-1))
x^3/(y - 1) - 3*x^2/(y - 1) + 3*x/(y - 1) - 1/(y - 1)
sage: expand((x+sin((x+y)^2))^2)
x^2 + 2*x*sin((x + y)^2) + sin((x + y)^2)^2

We can expand individual sides of a relation:

sage: a = (16*x-13)^2 == (3*x+5)^2/2
sage: a.expand()
256*x^2 - 416*x + 169 == 9/2*x^2 + 15*x + 25/2
sage: a.expand('left')
256*x^2 - 416*x + 169 == 1/2*(3*x + 5)^2
sage: a.expand('right')
(16*x - 13)^2 == 9/2*x^2 + 15*x + 25/2

TESTS:

sage: var('x,y')
(x, y)
sage: ((x + (2/3)*y)^3).expand()
x^3 + 2*x^2*y + 4/3*x*y^2 + 8/27*y^3
sage: expand( (x*sin(x) - cos(y)/x)^2 )
x^2*sin(x)^2 - 2*cos(y)*sin(x) + cos(y)^2/x^2
sage: f = (x-y)*(x+y); f
(x + y)*(x - y)
sage: f.expand()
x^2 - y^2

sage: a,b,c = var('a,b,c')
sage: x,y = var('x,y', domain='real')
sage: p,q = var('p,q', domain='positive')
sage: (c/2*(5*(3*a*b*x*y*p*q)^2)^(7/2*c)).expand()
1/2*45^(7/2*c)*(a^2*b^2*x^2*y^2)^(7/2*c)*c*p^(7*c)*q^(7*c)
sage: ((-(-a*x*p)^3*(b*y*p)^3)^(c/2)).expand()
(a^3*b^3*x^3*y^3)^(1/2*c)*p^(3*c)
sage: x,y,p,q = var('x,y,p,q', domain='complex')

rational_simplify(algorithm='full', map=False)

Simplify rational expressions.

INPUT:

• self - symbolic expression
• algorithm - (default: ‘full’) string which switches the algorithm for simplifications. Possible values are
  • ‘simple’ (simplify rational functions into quotient of two polynomials),
  • ‘full’ (apply repeatedly, if necessary)
  • ‘noexpand’ (convert to commmon denominator and add)
• map - (default: False) if True, the result is an expression whose leading operator is the same as that of the expression self but whose subparts are the results of applying simplification rules to the corresponding subparts of the expressions.

ALIAS: rational_simplify() and simplify_rational() are the same

DETAILS: We call Maxima functions ratsimp, fullratsimp and xthru. If each part of the expression has to be simplified separately, we use Maxima function map.

EXAMPLES:

sage: f = sin(x/(x^2 + x))
sage: f
sin(x/(x^2 + x))
sage: f.simplify_rational()
sin(1/(x + 1))

sage: f = ((x - 1)^(3/2) - (x + 1)*sqrt(x - 1))/sqrt((x - 1)*(x + 1)); f
-((x + 1)*sqrt(x - 1) - (x - 1)^(3/2))/sqrt((x + 1)*(x - 1))
sage: f.simplify_rational()
-2*sqrt(x - 1)/sqrt(x^2 - 1)

With map=True each term in a sum is simplified separately and thus the resuls are shorter for functions which are combination of rational and nonrational funtions. In the following example, we use this option if we want not to combine logarithm and the rational function into one fraction:

sage: f=(x^2-1)/(x+1)-ln(x)/(x+2)
sage: f.simplify_rational()
(x^2 + x - log(x) - 2)/(x + 2)
sage: f.simplify_rational(map=True)
x - log(x)/(x + 2) - 1

Here is an example from the Maxima documentation of where algorithm='simple' produces an (possibly useful) intermediate step:

sage: y = var('y')
sage: g = (x^(y/2) + 1)^2*(x^(y/2) - 1)^2/(x^y - 1)
sage: g.simplify_rational(algorithm='simple')
(x^(2*y) - 2*x^y + 1)/(x^y - 1)
sage: g.simplify_rational()
x^y - 1

With option algorithm='noexpand' we only convert to common denominators and add. No expansion of products is performed:

sage: f=1/(x+1)+x/(x+2)^2
sage: f.simplify_rational()
(2*x^2 + 5*x + 4)/(x^3 + 5*x^2 + 8*x + 4)
sage: f.simplify_rational(algorithm='noexpand')
((x + 2)^2 + (x + 1)*x)/((x + 2)^2*(x + 1))

real(hold=False)

Return the real part of this symbolic expression.

EXAMPLES:

sage: x = var('x')
sage: x.real_part()
real_part(x)
sage: SR(2+3*I).real_part()
2
sage: SR(CDF(2,3)).real_part()
2.0
sage: SR(CC(2,3)).real_part()
2.00000000000000

sage: f = log(x)
sage: f.real_part()
log(abs(x))

Using the hold parameter it is possible to prevent automatic evaluation:

sage: SR(2).real_part()
2
sage: SR(2).real_part(hold=True)
real_part(2)

This also works using functional notation:

sage: real_part(I,hold=True)
real_part(I)
sage: real_part(I)
0

To then evaluate again, we currently must use Maxima via simplify():

sage: a = SR(2).real_part(hold=True); a.simplify()
2

TESTS:

Check that trac ticket #12807 is fixed:

sage: (6*exp(i*pi/3)-6*exp(i*2*pi/3)).real_part()
6

real_part(hold=False)

Return the real part of this symbolic expression.

EXAMPLES:

sage: x = var('x')
sage: x.real_part()
real_part(x)
sage: SR(2+3*I).real_part()
2
sage: SR(CDF(2,3)).real_part()
2.0
sage: SR(CC(2,3)).real_part()
2.00000000000000

sage: f = log(x)
sage: f.real_part()
log(abs(x))

Using the hold parameter it is possible to prevent automatic evaluation:

sage: SR(2).real_part()
2
sage: SR(2).real_part(hold=True)
real_part(2)

This also works using functional notation:

sage: real_part(I,hold=True)
real_part(I)
sage: real_part(I)
0

To then evaluate again, we currently must use Maxima via simplify():

sage: a = SR(2).real_part(hold=True); a.simplify()
2

TESTS:

Check that trac ticket #12807 is fixed:

sage: (6*exp(i*pi/3)-6*exp(i*2*pi/3)).real_part()
6

rectform()

Convert this symbolic expression to rectangular form; that is, the form \(a + bi\) where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit.

Note

The name “rectangular” comes from the fact that, in the complex plane, \(a\) and \(bi\) are perpendicular.

INPUT:

• self – the expression to convert.

OUTPUT:

A new expression, equivalent to the original, but expressed in the form \(a + bi\).

ALGORITHM:

We call Maxima’s rectform() and return the result unmodified.

EXAMPLES:

The exponential form of \(\sin(x)\):

sage: f = (e^(I*x) - e^(-I*x)) / (2*I)
sage: f.rectform()
sin(x)

And \(\cos(x)\):

sage: f = (e^(I*x) + e^(-I*x)) / 2
sage: f.rectform()
cos(x)

In some cases, this will simplify the given expression. For example, here, \(e^{ik\pi}\), \(\sin(k\pi)=0\) should cancel leaving only \(\cos(k\pi)\) which can then be simplified:

sage: k = var('k')
sage: assume(k, 'integer')
sage: f = e^(I*pi*k)
sage: f.rectform()
(-1)^k

However, in general, the resulting expression may be more complicated than the original:

sage: f = e^(I*x)
sage: f.rectform()
cos(x) + I*sin(x)

TESTS:

If the expression is already in rectangular form, it should be left alone:

sage: a,b = var('a,b')
sage: assume((a, 'real'), (b, 'real'))
sage: f = a + b*I
sage: f.rectform()
a + I*b
sage: forget()

We can check with specific real numbers:

sage: a = RR.random_element()
sage: b = RR.random_element()
sage: f = a + b*I
sage: bool(f.rectform() == a + b*I)
True

If we decompose a complex number into its real and imaginary parts, they should correspond to the real and imaginary terms of the rectangular form:

sage: z = CC.random_element()
sage: a = z.real_part()
sage: b = z.imag_part()
sage: bool(SR(z).rectform() == a + b*I)
True

reduce_trig(var=None)

Combine products and powers of trigonometric and hyperbolic sin’s and cos’s of x into those of multiples of x. It also tries to eliminate these functions when they occur in denominators.

INPUT:

• self - a symbolic expression
• var - (default: None) the variable which is used for these transformations. If not specified, all variables are used.

OUTPUT:

A symbolic expression.

EXAMPLES:

sage: y=var('y')
sage: f=sin(x)*cos(x)^3+sin(y)^2
sage: f.reduce_trig()
-1/2*cos(2*y) + 1/8*sin(4*x) + 1/4*sin(2*x) + 1/2

To reduce only the expressions involving x we use optional parameter:

sage: f.reduce_trig(x)
sin(y)^2 + 1/8*sin(4*x) + 1/4*sin(2*x)

ALIASES: trig_reduce() and reduce_trig() are the same

residue(symbol)

Calculate the residue of self with respect to symbol.

INPUT:

• symbol - a symbolic variable or symbolic equality such as x == 5. If an equality is given, the expansion is around the value on the right hand side of the equality, otherwise at 0.

OUTPUT:

The residue of self.

Say, symbol is x == a, then this function calculates the residue of self at \(x=a\), i.e., the coefficient of \(1/(x-a)\) of the series expansion of self around \(a\).

EXAMPLES:

sage: (1/x).residue(x == 0)
1
sage: (1/x).residue(x == oo)
-1
sage: (1/x^2).residue(x == 0)
0
sage: (1/sin(x)).residue(x == 0)
1
sage: var('q, n, z')
(q, n, z)
sage: (-z^(-n-1)/(1-z/q)^2).residue(z == q).simplify_full()
(n + 1)/q^n
sage: var('s')
s
sage: zeta(s).residue(s == 1)
1

TESTS:

sage: (exp(x)/sin(x)^4).residue(x == 0)
5/6

Check that trac ticket #18372 is resolved:

sage: (1/(x^2 - x - 1)).residue(x == 1/2*sqrt(5) + 1/2)
1/5*sqrt(5)

rhs()

If self is a relational expression, return the right hand side of the relation. Otherwise, raise a ValueError.

EXAMPLES:

sage: x = var('x')
sage: eqn = (x-1)^2 <= x^2 - 2*x + 3
sage: eqn.right_hand_side()
x^2 - 2*x + 3
sage: eqn.rhs()
x^2 - 2*x + 3
sage: eqn.right()
x^2 - 2*x + 3

right()

If self is a relational expression, return the right hand side of the relation. Otherwise, raise a ValueError.

EXAMPLES:

sage: x = var('x')
sage: eqn = (x-1)^2 <= x^2 - 2*x + 3
sage: eqn.right_hand_side()
x^2 - 2*x + 3
sage: eqn.rhs()
x^2 - 2*x + 3
sage: eqn.right()
x^2 - 2*x + 3

right_hand_side()

If self is a relational expression, return the right hand side of the relation. Otherwise, raise a ValueError.

EXAMPLES:

sage: x = var('x')
sage: eqn = (x-1)^2 <= x^2 - 2*x + 3
sage: eqn.right_hand_side()
x^2 - 2*x + 3
sage: eqn.rhs()
x^2 - 2*x + 3
sage: eqn.right()
x^2 - 2*x + 3

roots(x=None, explicit_solutions=True, multiplicities=True, ring=None)

Return roots of self that can be found exactly, possibly with multiplicities. Not all roots are guaranteed to be found.

Warning

This is not a numerical solver - use find_root to solve for self == 0 numerically on an interval.

INPUT:

• x - variable to view the function in terms of (use default variable if not given)
• explicit_solutions - bool (default True); require that roots be explicit rather than implicit
• multiplicities - bool (default True); when True, return multiplicities
• ring - a ring (default None): if not None, convert self to a polynomial over ring and find roots over ring

OUTPUT:

A list of pairs (root, multiplicity) or list of roots.

If there are infinitely many roots, e.g., a function like \(\sin(x)\), only one is returned.

EXAMPLES:

sage: var('x, a')
(x, a)

A simple example:

sage: ((x^2-1)^2).roots()
[(-1, 2), (1, 2)]
sage: ((x^2-1)^2).roots(multiplicities=False)
[-1, 1]

A complicated example:

sage: f = expand((x^2 - 1)^3*(x^2 + 1)*(x-a)); f
-a*x^8 + x^9 + 2*a*x^6 - 2*x^7 - 2*a*x^2 + 2*x^3 + a - x

The default variable is \(a\), since it is the first in alphabetical order:

sage: f.roots()
[(x, 1)]

As a polynomial in \(a\), \(x\) is indeed a root:

sage: f.poly(a)
x^9 - 2*x^7 + 2*x^3 - (x^8 - 2*x^6 + 2*x^2 - 1)*a - x
sage: f(a=x)
0

The roots in terms of \(x\) are what we expect:

sage: f.roots(x)
[(a, 1), (-I, 1), (I, 1), (1, 3), (-1, 3)]

Only one root of \(\sin(x) = 0\) is given:

sage: f = sin(x)
sage: f.roots(x)
[(0, 1)]

Note

It is possible to solve a greater variety of equations using solve() and the keyword to_poly_solve, but only at the price of possibly encountering approximate solutions. See documentation for f.solve for more details.

We derive the roots of a general quadratic polynomial:

sage: var('a,b,c,x')
(a, b, c, x)
sage: (a*x^2 + b*x + c).roots(x)
[(-1/2*(b + sqrt(b^2 - 4*a*c))/a, 1), (-1/2*(b - sqrt(b^2 - 4*a*c))/a, 1)]

By default, all the roots are required to be explicit rather than implicit. To get implicit roots, pass explicit_solutions=False to .roots()

sage: var('x')
x
sage: f = x^(1/9) + (2^(8/9) - 2^(1/9))*(x - 1) - x^(8/9)
sage: f.roots()
Traceback (most recent call last):
...
RuntimeError: no explicit roots found
sage: f.roots(explicit_solutions=False)
[((2^(8/9) + x^(8/9) - 2^(1/9) - x^(1/9))/(2^(8/9) - 2^(1/9)), 1)]

Another example, but involving a degree 5 poly whose roots do not get computed explicitly:

sage: f = x^5 + x^3 + 17*x + 1
sage: f.roots()
Traceback (most recent call last):
...
RuntimeError: no explicit roots found
sage: f.roots(explicit_solutions=False)
[(x^5 + x^3 + 17*x + 1, 1)]
sage: f.roots(explicit_solutions=False, multiplicities=False)
[x^5 + x^3 + 17*x + 1]

Now let us find some roots over different rings:

sage: f.roots(ring=CC)
[(-0.0588115223184..., 1), (-1.331099917875... - 1.52241655183732*I, 1), (-1.331099917875... + 1.52241655183732*I, 1), (1.36050567903502 - 1.51880872209965*I, 1), (1.36050567903502 + 1.51880872209965*I, 1)]
sage: (2.5*f).roots(ring=RR)
[(-0.058811522318449..., 1)]
sage: f.roots(ring=CC, multiplicities=False)
[-0.05881152231844..., -1.331099917875... - 1.52241655183732*I, -1.331099917875... + 1.52241655183732*I, 1.36050567903502 - 1.51880872209965*I, 1.36050567903502 + 1.51880872209965*I]
sage: f.roots(ring=QQ)
[]
sage: f.roots(ring=QQbar, multiplicities=False)
[-0.05881152231844944?, -1.331099917875796? - 1.522416551837318?*I, -1.331099917875796? + 1.522416551837318?*I, 1.360505679035020? - 1.518808722099650?*I, 1.360505679035020? + 1.518808722099650?*I]

Root finding over finite fields:

sage: f.roots(ring=GF(7^2, 'a'))
[(3, 1), (4*a + 6, 2), (3*a + 3, 2)]

TESTS:

sage: (sqrt(3) * f).roots(ring=QQ)
Traceback (most recent call last):
...
TypeError: unable to convert sqrt(3) to a rational

Check if trac ticket #9538 is fixed:

sage: var('f6,f5,f4,x')
(f6, f5, f4, x)
sage: e=15*f6*x^2 + 5*f5*x + f4
sage: res = e.roots(x); res
[(-1/30*(5*f5 + sqrt(25*f5^2 - 60*f4*f6))/f6, 1), (-1/30*(5*f5 - sqrt(25*f5^2 - 60*f4*f6))/f6, 1)]
sage: e.subs(x=res[0][0]).is_zero()
True

round()

Round this expression to the nearest integer.

EXAMPLES:

sage: u = sqrt(43203735824841025516773866131535024)
sage: u.round()
207855083711803945
sage: t = sqrt(Integer('1'*1000)).round(); print str(t)[-10:]
3333333333
sage: (-sqrt(110)).round()
-10
sage: (-sqrt(115)).round()
-11
sage: (sqrt(-3)).round()
Traceback (most recent call last):
...
ValueError: could not convert sqrt(-3) to a real number

series(symbol, order=None)

Return the power series expansion of self in terms of the given variable to the given order.

INPUT:

• symbol - a symbolic variable or symbolic equality such as x == 5; if an equality is given, the expansion is around the value on the right hand side of the equality
• order - an integer; if nothing given, it is set to the global default (20), which can be changed using set_series_precision()

OUTPUT:

A power series.

To truncate the power series and obtain a normal expression, use the truncate() command.

EXAMPLES:

We expand a polynomial in \(x\) about 0, about \(1\), and also truncate it back to a polynomial:

sage: var('x,y')
(x, y)
sage: f = (x^3 - sin(y)*x^2 - 5*x + 3); f
x^3 - x^2*sin(y) - 5*x + 3
sage: g = f.series(x, 4); g
3 + (-5)*x + (-sin(y))*x^2 + 1*x^3
sage: g.truncate()
x^3 - x^2*sin(y) - 5*x + 3
sage: g = f.series(x==1, 4); g
(-sin(y) - 1) + (-2*sin(y) - 2)*(x - 1) + (-sin(y) + 3)*(x - 1)^2 + 1*(x - 1)^3
sage: h = g.truncate(); h
(x - 1)^3 - (x - 1)^2*(sin(y) - 3) - 2*(x - 1)*(sin(y) + 1) - sin(y) - 1
sage: h.expand()
x^3 - x^2*sin(y) - 5*x + 3

We computer another series expansion of an analytic function:

sage: f = sin(x)/x^2
sage: f.series(x,7)
1*x^(-1) + (-1/6)*x + 1/120*x^3 + (-1/5040)*x^5 + Order(x^7)
sage: f.series(x)
1*x^(-1) + (-1/6)*x + ... + Order(x^20)
sage: f.series(x==1,3)
(sin(1)) + (cos(1) - 2*sin(1))*(x - 1) + (-2*cos(1) + 5/2*sin(1))*(x - 1)^2 + Order((x - 1)^3)
sage: f.series(x==1,3).truncate().expand()
-2*x^2*cos(1) + 5/2*x^2*sin(1) + 5*x*cos(1) - 7*x*sin(1) - 3*cos(1) + 11/2*sin(1)

Expressions formed by combining series can be expanded by applying series again:

sage: (1/(1-x)).series(x, 3)+(1/(1+x)).series(x,3)
(1 + (-1)*x + 1*x^2 + Order(x^3)) + (1 + 1*x + 1*x^2 + Order(x^3))
sage: _.series(x,3)
2 + 2*x^2 + Order(x^3)
sage: (1/(1-x)).series(x, 3)*(1/(1+x)).series(x,3)
(1 + (-1)*x + 1*x^2 + Order(x^3))*(1 + 1*x + 1*x^2 + Order(x^3))
sage: _.series(x,3)
1 + 1*x^2 + Order(x^3)

Following the GiNaC tutorial, we use John Machin’s amazing formula \(\pi = 16 \tan^{-1}(1/5) - 4 \tan^{-1}(1/239)\) to compute digits of \(\pi\). We expand the arc tangent around 0 and insert the fractions 1/5 and 1/239.

sage: x = var('x')
sage: f = atan(x).series(x, 10); f
1*x + (-1/3)*x^3 + 1/5*x^5 + (-1/7)*x^7 + 1/9*x^9 + Order(x^10)
sage: float(16*f.subs(x==1/5) - 4*f.subs(x==1/239))
3.1415926824043994

TESTS:

Check if trac ticket #8943 is fixed:

sage: ((1+arctan(x))**(1/x)).series(x==0, 3)
(e) + (-1/2*e)*x + (1/8*e)*x^2 + Order(x^3)

Order may be negative:

sage: f = sin(x)^(-2); f.series(x, -1)
1*x^(-2) + Order(1/x)

Check if changing global series precision does it right:

sage: set_series_precision(3)
sage: (1/(1-2*x)).series(x)
1 + 2*x + 4*x^2 + Order(x^3)
sage: set_series_precision(20)

show()

Pretty-Print this symbolic expression

This typeset it nicely and prints it immediately.

OUTPUT:

This method does not return anything. Like print, output is sent directly to the screen.

EXAMPLES:

sage: (x^2 + 1).show()
<html><script type="math/tex">\newcommand{\Bold}[1]{\mathbf{#1}}x^{2} + 1</script></html>

simplify()

Return a simplified version of this symbolic expression.

Note

Currently, this just sends the expression to Maxima and converts it back to Sage.

See also

simplify_full(), simplify_trig(), simplify_rational(), simplify_rectform() simplify_factorial(), simplify_log(), simplify_real(), simplify_hypergeometric(), canonicalize_radical()

EXAMPLES:

sage: a = var('a'); f = x*sin(2)/(x^a); f
x*sin(2)/x^a
sage: f.simplify()
x^(-a + 1)*sin(2)

TESTS:

Check that trac ticket #14637 is fixed:

sage: assume(x > 0, x < pi/2)
sage: acos(cos(x)).simplify()
x
sage: forget()

simplify_exp(*args, **kwds)

Deprecated: Use canonicalize_radical() instead. See trac ticket #11912 for details.

simplify_factorial()

Simplify by combining expressions with factorials, and by expanding binomials into factorials.

ALIAS: factorial_simplify and simplify_factorial are the same

EXAMPLES:

Some examples are relatively clear:

sage: var('n,k')
(n, k)
sage: f = factorial(n+1)/factorial(n); f
factorial(n + 1)/factorial(n)
sage: f.simplify_factorial()
n + 1

sage: f = factorial(n)*(n+1); f
(n + 1)*factorial(n)
sage: simplify(f)
(n + 1)*factorial(n)
sage: f.simplify_factorial()
factorial(n + 1)

sage: f = binomial(n, k)*factorial(k)*factorial(n-k); f
binomial(n, k)*factorial(k)*factorial(-k + n)
sage: f.simplify_factorial()
factorial(n)

A more complicated example, which needs further processing:

sage: f = factorial(x)/factorial(x-2)/2 + factorial(x+1)/factorial(x)/2; f
1/2*factorial(x + 1)/factorial(x) + 1/2*factorial(x)/factorial(x - 2)
sage: g = f.simplify_factorial(); g
1/2*(x - 1)*x + 1/2*x + 1/2
sage: g.simplify_rational()
1/2*x^2 + 1/2

TESTS:

Check that the problem with applying \(full_simplify()\) to gamma functions (trac ticket #9240) has been fixed:

sage: gamma(1/3)
gamma(1/3)
sage: gamma(1/3).full_simplify()
gamma(1/3)
sage: gamma(4/3)
gamma(4/3)
sage: gamma(4/3).full_simplify()
1/3*gamma(1/3)

simplify_full()

Apply simplify_factorial(), simplify_rectform(), simplify_trig(), simplify_rational(), and then expand_sum() to self (in that order).

ALIAS: simplify_full and full_simplify are the same.

EXAMPLES:

sage: f = sin(x)^2 + cos(x)^2
sage: f.simplify_full()
1

sage: f = sin(x/(x^2 + x))
sage: f.simplify_full()
sin(1/(x + 1))

sage: var('n,k')
(n, k)
sage: f = binomial(n,k)*factorial(k)*factorial(n-k)
sage: f.simplify_full()
factorial(n)

TESTS:

There are two square roots of \($(x + 1)^2$\), so this should not be simplified to \($x + 1$\), trac ticket #12737:

sage: f = sqrt((x + 1)^2)
sage: f.simplify_full()
sqrt(x^2 + 2*x + 1)

The imaginary part of an expression should not change under simplification; trac ticket #11934:

sage: f = sqrt(-8*(4*sqrt(2) - 7)*x^4 + 16*(3*sqrt(2) - 5)*x^3)
sage: original = f.imag_part()
sage: simplified = f.full_simplify().imag_part()
sage: original - simplified
0

The invalid simplification from trac ticket #12322 should not occur after trac ticket #12737:

sage: t = var('t')
sage: assume(t, 'complex')
sage: assumptions()
[t is complex]
sage: f = (1/2)*log(2*t) + (1/2)*log(1/t)
sage: f.simplify_full()
1/2*log(2*t) - 1/2*log(t)
sage: forget()

Complex logs are not contracted, trac ticket #17556:

sage: x,y = SR.var('x,y')
sage: assume(y, 'complex')
sage: f = log(x*y) - (log(x) + log(y))
sage: f.simplify_full()
log(x*y) - log(x) - log(y)
sage: forget()

The simplifications from simplify_rectform() are performed, trac ticket #17556:

sage: f = ( e^(I*x) - e^(-I*x) ) / ( I*e^(I*x) + I*e^(-I*x) )
sage: f.simplify_full()
sin(x)/cos(x)

simplify_hypergeometric(algorithm='maxima')

Simplify an expression containing hypergeometric functions.

INPUT:

• algorithm – (default: 'maxima') the algorithm to use for for simplification. Implemented are 'maxima', which uses Maxima’s hgfred function, and 'sage', which uses an algorithm implemented in the hypergeometric module

ALIAS: hypergeometric_simplify() and simplify_hypergeometric() are the same

EXAMPLES:

sage: hypergeometric((5, 4), (4, 1, 2, 3),
....:                x).simplify_hypergeometric()
1/144*x^2*hypergeometric((), (3, 4), x) +...
1/3*x*hypergeometric((), (2, 3), x) + hypergeometric((), (1, 2), x)
sage: (2*hypergeometric((), (), x)).simplify_hypergeometric()
2*e^x
sage: (nest(lambda y: hypergeometric([y], [1], x), 3, 1)
....:  .simplify_hypergeometric())
laguerre(-laguerre(-e^x, x), x)
sage: (nest(lambda y: hypergeometric([y], [1], x), 3, 1)
....:  .simplify_hypergeometric(algorithm='sage'))
hypergeometric((hypergeometric((e^x,), (1,), x),), (1,), x)

simplify_log(algorithm=None)

Simplify a (real) symbolic expression that contains logarithms.

The given expression is scanned recursively, transforming subexpressions of the form \(a \log(b) + c \log(d)\) into \(\log(b^{a} d^{c})\) before simplifying within the log().

The user can specify conditions that \(a\) and \(c\) must satisfy before this transformation will be performed using the optional parameter algorithm.

Warning

This is only safe to call if every variable in the given expression is assumed to be real. The simplification it performs is in general not valid over the complex numbers. For example:

sage: x,y = SR.var('x,y')
sage: f = log(x*y) - (log(x) + log(y))
sage: f(x=-1, y=i)
-2*I*pi
sage: f.simplify_log()
0

INPUT:

• self - expression to be simplified
• algorithm - (default: None) optional, governs the condition on \(a\) and \(c\) which must be satisfied to contract expression \(a \log(b) + c \log(d)\). Values are
  • None (use Maxima default, integers),
  • 'one' (1 and -1),
  • 'ratios' (rational numbers),
  • 'constants' (constants),
  • 'all' (all expressions).

ALGORITHM:

This uses the Maxima logcontract() command.

ALIAS:

log_simplify() and simplify_log() are the same.

EXAMPLES:

sage: x,y,t=var('x y t')

Only two first terms are contracted in the following example; the logarithm with coefficient \(\frac{1}{2}\) is not contracted:

sage: f = log(x)+2*log(y)+1/2*log(t)
sage: f.simplify_log()
log(x*y^2) + 1/2*log(t)

To contract all terms in the previous example, we use the 'ratios' algorithm:

sage: f.simplify_log(algorithm='ratios')
log(sqrt(t)*x*y^2)

To contract terms with no coefficient (more precisely, with coefficients \(1\) and \(-1\)), we use the 'one' algorithm:

sage: f = log(x)+2*log(y)-log(t)
sage: f.simplify_log('one')
2*log(y) + log(x/t)

sage: f = log(x)+log(y)-1/3*log((x+1))
sage: f.simplify_log()
log(x*y) - 1/3*log(x + 1)

sage: f.simplify_log('ratios')
log(x*y/(x + 1)^(1/3))

\(\pi\) is an irrational number; to contract logarithms in the following example we have to set algorithm to 'constants' or 'all':

sage: f = log(x)+log(y)-pi*log((x+1))
sage: f.simplify_log('constants')
log(x*y/(x + 1)^pi)

x*log(9) is contracted only if algorithm is 'all':

sage: (x*log(9)).simplify_log()
x*log(9)
sage: (x*log(9)).simplify_log('all')
log(9^x)

TESTS:

Ensure that the option algorithm from one call has no influence upon future calls (a Maxima flag was set, and we have to ensure that its value has been restored):

sage: f = log(x)+2*log(y)+1/2*log(t)
sage: f.simplify_log('one')
1/2*log(t) + log(x) + 2*log(y)

sage: f.simplify_log('ratios')
log(sqrt(t)*x*y^2)

sage: f.simplify_log()
log(x*y^2) + 1/2*log(t)

This shows that the issue at trac ticket #7334 is fixed. Maxima intentionally keeps the expression inside the log factored:

sage: log_expr = (log(sqrt(2)-1)+log(sqrt(2)+1))
sage: log_expr.simplify_log('all')
log((sqrt(2) + 1)*(sqrt(2) - 1))
sage: _.simplify_rational()
0

We should use the current simplification domain rather than set it to ‘real’ explicitly (trac ticket #12780):

sage: f = sqrt(x^2)
sage: f.simplify_log()
sqrt(x^2)
sage: from sage.calculus.calculus import maxima
sage: maxima('domain: real;')
real
sage: f.simplify_log()
abs(x)
sage: maxima('domain: complex;')
complex

AUTHORS:

• Robert Marik (11-2009)

simplify_radical(*args, **kwds)

Deprecated: Use canonicalize_radical() instead. See trac ticket #11912 for details.

simplify_rational(algorithm='full', map=False)

Simplify rational expressions.

INPUT:

• self - symbolic expression
• algorithm - (default: ‘full’) string which switches the algorithm for simplifications. Possible values are
  • ‘simple’ (simplify rational functions into quotient of two polynomials),
  • ‘full’ (apply repeatedly, if necessary)
  • ‘noexpand’ (convert to commmon denominator and add)
• map - (default: False) if True, the result is an expression whose leading operator is the same as that of the expression self but whose subparts are the results of applying simplification rules to the corresponding subparts of the expressions.

ALIAS: rational_simplify() and simplify_rational() are the same

DETAILS: We call Maxima functions ratsimp, fullratsimp and xthru. If each part of the expression has to be simplified separately, we use Maxima function map.

EXAMPLES:

sage: f = sin(x/(x^2 + x))
sage: f
sin(x/(x^2 + x))
sage: f.simplify_rational()
sin(1/(x + 1))

sage: f = ((x - 1)^(3/2) - (x + 1)*sqrt(x - 1))/sqrt((x - 1)*(x + 1)); f
-((x + 1)*sqrt(x - 1) - (x - 1)^(3/2))/sqrt((x + 1)*(x - 1))
sage: f.simplify_rational()
-2*sqrt(x - 1)/sqrt(x^2 - 1)

With map=True each term in a sum is simplified separately and thus the resuls are shorter for functions which are combination of rational and nonrational funtions. In the following example, we use this option if we want not to combine logarithm and the rational function into one fraction:

sage: f=(x^2-1)/(x+1)-ln(x)/(x+2)
sage: f.simplify_rational()
(x^2 + x - log(x) - 2)/(x + 2)
sage: f.simplify_rational(map=True)
x - log(x)/(x + 2) - 1

Here is an example from the Maxima documentation of where algorithm='simple' produces an (possibly useful) intermediate step:

sage: y = var('y')
sage: g = (x^(y/2) + 1)^2*(x^(y/2) - 1)^2/(x^y - 1)
sage: g.simplify_rational(algorithm='simple')
(x^(2*y) - 2*x^y + 1)/(x^y - 1)
sage: g.simplify_rational()
x^y - 1

With option algorithm='noexpand' we only convert to common denominators and add. No expansion of products is performed:

sage: f=1/(x+1)+x/(x+2)^2
sage: f.simplify_rational()
(2*x^2 + 5*x + 4)/(x^3 + 5*x^2 + 8*x + 4)
sage: f.simplify_rational(algorithm='noexpand')
((x + 2)^2 + (x + 1)*x)/((x + 2)^2*(x + 1))

simplify_real()

Simplify the given expression over the real numbers. This allows the simplification of \(\sqrt{x^{2}}\) into \(\left|x\right|\) and the contraction of \(\log(x) + \log(y)\) into \(\log(xy)\).

INPUT:

• self – the expression to convert.

OUTPUT:

A new expression, equivalent to the original one under the assumption that the variables involved are real.

EXAMPLES:

sage: f = sqrt(x^2)
sage: f.simplify_real()
abs(x)

sage: y = SR.var('y')
sage: f = log(x) + 2*log(y)
sage: f.simplify_real()
log(x*y^2)

TESTS:

We set the Maxima domain variable to ‘real’ before we call out to Maxima. When we return, however, we should set the domain back to what it was, rather than assuming that it was ‘complex’:

sage: from sage.calculus.calculus import maxima
sage: maxima('domain: real;')
real
sage: x.simplify_real()
x
sage: maxima('domain;')
real
sage: maxima('domain: complex;')
complex

We forget the assumptions that our variables are real after simplification; make sure we don’t forget an assumption that existed before we were called:

sage: assume(x, 'real')
sage: x.simplify_real()
x
sage: assumptions()
[x is real]
sage: forget()

We also want to be sure that we don’t forget assumptions on other variables:

sage: x,y,z = SR.var('x,y,z')
sage: assume(y, 'integer')
sage: assume(z, 'antisymmetric')
sage: x.simplify_real()
x
sage: assumptions()
[y is integer, z is antisymmetric]
sage: forget()

No new assumptions should exist after the call:

sage: assumptions()
[]
sage: x.simplify_real()
x
sage: assumptions()
[]

simplify_rectform(complexity_measure='string_length')

Attempt to simplify this expression by expressing it in the form \(a + bi\) where both \(a\) and \(b\) are real. This transformation is generally not a simplification, so we use the given complexity_measure to discard non-simplifications.

INPUT:

• self – the expression to simplify.
• complexity_measure – (default: sage.symbolic.complexity_measures.string_length) a function taking a symbolic expression as an argument and returning a measure of that expressions complexity. If None is supplied, the simplification will be performed regardless of the result.

OUTPUT:

If the transformation produces a simpler expression (according to complexity_measure) then that simpler expression is returned. Otherwise, the original expression is returned.

ALGORITHM:

We first call rectform() on the given expression. Then, the supplied complexity measure is used to determine whether or not the result is simpler than the original expression.

EXAMPLES:

The exponential form of \(\tan(x)\):

sage: f = ( e^(I*x) - e^(-I*x) ) / ( I*e^(I*x) + I*e^(-I*x) )
sage: f.simplify_rectform()
sin(x)/cos(x)

This should not be expanded with Euler’s formula since the resulting expression is longer when considered as a string, and the default complexity_measure uses string length to determine which expression is simpler:

sage: f = e^(I*x)
sage: f.simplify_rectform()
e^(I*x)

However, if we pass None as our complexity measure, it is:

sage: f = e^(I*x)
sage: f.simplify_rectform(complexity_measure = None)
cos(x) + I*sin(x)

TESTS:

When given None, we should always call rectform() and return the result:

sage: polynomials = QQ['x']
sage: f = SR(polynomials.random_element())
sage: g = f.simplify_rectform(complexity_measure = None)
sage: bool(g == f.rectform())
True

simplify_trig(expand=True)

Optionally expand and then employ identities such as \(\sin(x)^2 + \cos(x)^2 = 1\), \(\cosh(x)^2 - \sinh(x)^2 = 1\), \(\sin(x)\csc(x) = 1\), or \(\tanh(x)=\sinh(x)/\cosh(x)\) to simplify expressions containing tan, sec, etc., to sin, cos, sinh, cosh.

INPUT:

• self - symbolic expression
• expand - (default:True) if True, expands trigonometric and hyperbolic functions of sums of angles and of multiple angles occurring in self first. For best results, self should be expanded. See also expand_trig() to get more controls on this expansion.

ALIAS: trig_simplify() and simplify_trig() are the same

EXAMPLES:

sage: f = sin(x)^2 + cos(x)^2; f
cos(x)^2 + sin(x)^2
sage: f.simplify()
cos(x)^2 + sin(x)^2
sage: f.simplify_trig()
1
sage: h = sin(x)*csc(x)
sage: h.simplify_trig()
1
sage: k = tanh(x)*cosh(2*x)
sage: k.simplify_trig()
(2*sinh(x)^3 + sinh(x))/cosh(x)

In some cases we do not want to expand:

sage: f=tan(3*x)
sage: f.simplify_trig()
(4*cos(x)^2 - 1)*sin(x)/(4*cos(x)^3 - 3*cos(x))
sage: f.simplify_trig(False)
sin(3*x)/cos(3*x)

sin(hold=False)

EXAMPLES:

sage: var('x, y')
(x, y)
sage: sin(x^2 + y^2)
sin(x^2 + y^2)
sage: sin(sage.symbolic.constants.pi)
0
sage: sin(SR(1))
sin(1)
sage: sin(SR(RealField(150)(1)))
0.84147098480789650665250232163029899962256306

Using the hold parameter it is possible to prevent automatic evaluation:

sage: SR(0).sin()
0
sage: SR(0).sin(hold=True)
sin(0)

This also works using functional notation:

sage: sin(0,hold=True)
sin(0)
sage: sin(0)
0

To then evaluate again, we currently must use Maxima via simplify():

sage: a = SR(0).sin(hold=True); a.simplify()
0

TESTS:

sage: SR(oo).sin()
Traceback (most recent call last):
...
RuntimeError: sin_eval(): sin(infinity) encountered
sage: SR(-oo).sin()
Traceback (most recent call last):
...
RuntimeError: sin_eval(): sin(infinity) encountered
sage: SR(unsigned_infinity).sin()
Traceback (most recent call last):
...
RuntimeError: sin_eval(): sin(infinity) encountered

sinh(hold=False)

Return sinh of self.

We have \(\sinh(x) = (e^{x} - e^{-x})/2\).

EXAMPLES:

sage: x.sinh()
sinh(x)
sage: SR(1).sinh()
sinh(1)
sage: SR(0).sinh()
0
sage: SR(1.0).sinh()
1.17520119364380
sage: maxima('sinh(1.0)')
1.17520119364380...

sinh(1.0000000000000000000000000)
sage: SR(1).sinh().n(90)
1.1752011936438014568823819
sage: SR(RIF(1)).sinh()
1.175201193643802?

To prevent automatic evaluation use the hold argument:

sage: arccosh(x).sinh()
sqrt(x + 1)*sqrt(x - 1)
sage: arccosh(x).sinh(hold=True)
sinh(arccosh(x))

This also works using functional notation:

sage: sinh(arccosh(x),hold=True)
sinh(arccosh(x))
sage: sinh(arccosh(x))
sqrt(x + 1)*sqrt(x - 1)

To then evaluate again, we currently must use Maxima via simplify():

sage: a = arccosh(x).sinh(hold=True); a.simplify()
sqrt(x + 1)*sqrt(x - 1)

TESTS:

sage: SR(oo).sinh()
+Infinity
sage: SR(-oo).sinh()
-Infinity
sage: SR(unsigned_infinity).sinh()
Traceback (most recent call last):
...
RuntimeError: sinh_eval(): sinh(unsigned_infinity) encountered

solve(x, multiplicities=False, solution_dict=False, explicit_solutions=False, to_poly_solve=False)

Analytically solve the equation self == 0 or a univariate inequality for the variable \(x\).

Warning

This is not a numerical solver - use find_root to solve for self == 0 numerically on an interval.

INPUT:

• x - variable(s) to solve for
• multiplicities - bool (default: False); if True, return corresponding multiplicities. This keyword is incompatible with to_poly_solve=True and does not make any sense when solving an inequality.
• solution_dict - bool (default: False); if True or non-zero, return a list of dictionaries containing solutions. Not used when solving an inequality.
• explicit_solutions - bool (default: False); require that all roots be explicit rather than implicit. Not used when solving an inequality.
• to_poly_solve - bool (default: False) or string; use Maxima’s to_poly_solver package to search for more possible solutions, but possibly encounter approximate solutions. This keyword is incompatible with multiplicities=True and is not used when solving an inequality. Setting to_poly_solve to 'force' omits Maxima’s solve command (useful when some solutions of trigonometric equations are lost).

EXAMPLES:

sage: z = var('z')
sage: (z^5 - 1).solve(z)
[z == e^(2/5*I*pi), z == e^(4/5*I*pi), z == e^(-4/5*I*pi), z == e^(-2/5*I*pi), z == 1]

sage: solve((z^3-1)^3, z, multiplicities=True)
([z == 1/2*I*sqrt(3) - 1/2, z == -1/2*I*sqrt(3) - 1/2, z == 1], [3, 3, 3])

A simple example to show the use of the keyword multiplicities:

sage: ((x^2-1)^2).solve(x)
[x == -1, x == 1]
sage: ((x^2-1)^2).solve(x,multiplicities=True)
([x == -1, x == 1], [2, 2])
sage: ((x^2-1)^2).solve(x,multiplicities=True,to_poly_solve=True)
Traceback (most recent call last):
...
NotImplementedError: to_poly_solve does not return multiplicities

Here is how the explicit_solutions keyword functions:

sage: solve(sin(x)==x,x)
[x == sin(x)]
sage: solve(sin(x)==x,x,explicit_solutions=True)
[]
sage: solve(x*sin(x)==x^2,x)
[x == 0, x == sin(x)]
sage: solve(x*sin(x)==x^2,x,explicit_solutions=True)
[x == 0]

The following examples show the use of the keyword to_poly_solve:

sage: solve(abs(1-abs(1-x)) == 10, x)
[abs(abs(x - 1) - 1) == 10]
sage: solve(abs(1-abs(1-x)) == 10, x, to_poly_solve=True)
[x == -10, x == 12]

sage: var('Q')
Q
sage: solve(Q*sqrt(Q^2 + 2) - 1, Q)
[Q == 1/sqrt(Q^2 + 2)]
sage: solve(Q*sqrt(Q^2 + 2) - 1, Q, to_poly_solve=True)
[Q == 1/sqrt(-sqrt(2) + 1), Q == 1/sqrt(sqrt(2) + 1)]

In some cases there may be infinitely many solutions indexed by a dummy variable. If it begins with z, it is implicitly assumed to be an integer, a real if with r, and so on:

sage: solve( sin(x)==cos(x), x, to_poly_solve=True)
[x == 1/4*pi + pi*z...]

An effort is made to only return solutions that satisfy the current assumptions:

sage: solve(x^2==4, x)
[x == -2, x == 2]
sage: assume(x<0)
sage: solve(x^2==4, x)
[x == -2]
sage: solve((x^2-4)^2 == 0, x, multiplicities=True)
([x == -2], [2])
sage: solve(x^2==2, x)
[x == -sqrt(2)]
sage: assume(x, 'rational')
sage: solve(x^2 == 2, x)
[]
sage: solve(x^2==2-z, x)
[x == -sqrt(-z + 2)]
sage: solve((x-z)^2==2, x)
[x == z - sqrt(2), x == z + sqrt(2)]

In some cases it may be worthwhile to directly use to_poly_solve if one suspects some answers are being missed:

sage: forget()
sage: solve(cos(x)==0, x)
[x == 1/2*pi]
sage: solve(cos(x)==0, x, to_poly_solve=True)
[x == 1/2*pi]
sage: solve(cos(x)==0, x, to_poly_solve='force')
[x == 1/2*pi + pi*z77]

The same may also apply if a returned unsolved expression has a denominator, but the original one did not:

sage: solve(cos(x) * sin(x) == 1/2, x, to_poly_solve=True)
[sin(x) == 1/2/cos(x)]
sage: solve(cos(x) * sin(x) == 1/2, x, to_poly_solve=True, explicit_solutions=True)
[x == 1/4*pi + pi*z...]
sage: solve(cos(x) * sin(x) == 1/2, x, to_poly_solve='force')
[x == 1/4*pi + pi*z...]

We can also solve for several variables:

sage: var('b, c')
(b, c)
sage: solve((b-1)*(c-1), [b,c])
[[b == 1, c == r4], [b == r5, c == 1]]

We use sympy for Diophantine equations, see solve_diophantine()

sage: assume(x, 'integer')
sage: assume(z, 'integer')
sage: solve((x-z)^2==2, x)
[]

sage: forget()

Some basic inequalities can be also solved:

sage: x,y=var('x,y'); (ln(x)-ln(y)>0).solve(x)
[[log(x) - log(y) > 0]]

sage: x,y=var('x,y'); (ln(x)>ln(y)).solve(x)  # random
[[0 < y, y < x, 0 < x]]
[[y < x, 0 < y]]

TESTS:

trac ticket #7325 (solving inequalities):

sage: (x^2>1).solve(x)
[[x < -1], [x > 1]]

Catch error message from Maxima:

sage: solve(acot(x),x)
[]

sage: solve(acot(x),x,to_poly_solve=True)
[]

trac ticket #7491 fixed:

sage: y=var('y')
sage: solve(y==y,y)
[y == r1]
sage: solve(y==y,y,multiplicities=True)
([y == r1], [])

sage: from sage.symbolic.assumptions import GenericDeclaration
sage: GenericDeclaration(x, 'rational').assume()
sage: solve(x^2 == 2, x)
[]
sage: forget()

trac ticket #8390 fixed:

sage: solve(sin(x)==1/2,x)
[x == 1/6*pi]

sage: solve(sin(x)==1/2,x,to_poly_solve=True)
[x == 1/6*pi]

sage: solve(sin(x)==1/2, x, to_poly_solve='force')
[x == 1/6*pi + 2*pi*z..., x == 5/6*pi + 2*pi*z...]

trac ticket #11618 fixed:

sage: g(x)=0
sage: solve(g(x)==0,x,solution_dict=True)
[{x: r1}]

trac ticket #13286 fixed:

sage: solve([x-4], [x])
[x == 4]

trac ticket #13645: fixed:

sage: x.solve((1,2))
Traceback (most recent call last):
...
TypeError: (1, 2) are not valid variables.

trac ticket #17128: fixed:

sage: var('x,y')
(x, y)
sage: f = x+y
sage: sol = f.solve([x, y], solution_dict=True)
sage: sol[0].get(x) + sol[0].get(y)
0

trac ticket #16651 fixed:

sage: (x^7-x-1).solve(x, to_poly_solve=True)     # abs tol 1e-6
[x == 1.11277569705,
 x == (-0.363623519329 - 0.952561195261*I),
 x == (0.617093477784 - 0.900864951949*I),
 x == (-0.809857800594 - 0.262869645851*I),
 x == (-0.809857800594 + 0.262869645851*I),
 x == (0.617093477784 + 0.900864951949*I),
 x == (-0.363623519329 + 0.952561195261*I)]

solve_diophantine(x=None, solution_dict=False)

Solve a polynomial equation in the integers (a so called Diophantine).

If the argument is just a polynomial expression, equate to zero. If solution_dict=True return a list of dictionaries instead of a list of tuples.

EXAMPLES:

sage: x,y = var('x,y')
sage: solve_diophantine(3*x == 4)
[]
sage: solve_diophantine(x^2 - 9)
[-3, 3]
sage: sorted(solve_diophantine(x^2 + y^2 == 25))
[(-4, -3), (-4, 3), (0, -5), (0, 5), (4, -3), (4, 3)]

The function is used when solve() is called with all variables assumed integer:

sage: assume(x, 'integer')
sage: assume(y, 'integer')
sage: sorted(solve(x*y == 1, (x,y)))
[(-1, -1), (1, 1)]

You can also pick specific variables, and get the solution as a dictionary:

sage: solve_diophantine(x*y == 10, x)
[-10, -5, -2, -1, 1, 2, 5, 10]
sage: sorted(solve_diophantine(x*y - y == 10, (x,y)))
[(-9, -1), (-4, -2), (-1, -5), (0, -10), (2, 10), (3, 5), (6, 2), (11, 1)]
sage: res = solve_diophantine(x*y - y == 10, solution_dict=True)
sage: sol = [{y: -5, x: -1}, {y: -10, x: 0}, {y: -1, x: -9}, {y: -2, x: -4}, {y: 10, x: 2}, {y: 1, x: 11}, {y: 2, x: 6}, {y: 5, x: 3}]
sage: all(solution in res for solution in sol) and bool(len(res) == len(sol))
True

If the solution is parametrized the parameter(s) are not defined, but you can substitute them with specific integer values:

sage: x,y,z = var('x,y,z')
sage: sol=solve_diophantine(x^2-y==0); sol
(t, t^2)
sage: print [(sol[0].subs(t=t),sol[1].subs(t=t)) for t in range(-3,4)]
[(-3, 9), (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4), (3, 9)]
sage: sol = solve_diophantine(x^2 + y^2 == z^2); sol
(2*p*q, p^2 - q^2, p^2 + q^2)
sage: print [(sol[0].subs(p=p,q=q),sol[1].subs(p=p,q=q),sol[2].subs(p=p,q=q)) for p in range(1,4) for q in range(1,4)]
[(2, 0, 2), (4, -3, 5), (6, -8, 10), (4, 3, 5), (8, 0, 8), (12, -5, 13), (6, 8, 10), (12, 5, 13), (18, 0, 18)]

Solve Brahmagupta-Pell equations:

sage: sol = solve_diophantine(x^2 - 2*y^2 == 1); sol
(sqrt(2)*(2*sqrt(2) + 3)^t - sqrt(2)*(-2*sqrt(2) + 3)^t + 3/2*(2*sqrt(2) + 3)^t + 3/2*(-2*sqrt(2) + 3)^t,
 3/4*sqrt(2)*(2*sqrt(2) + 3)^t - 3/4*sqrt(2)*(-2*sqrt(2) + 3)^t + (2*sqrt(2) + 3)^t + (-2*sqrt(2) + 3)^t)
sage: print [(sol[0].subs(t=t).simplify_full(),sol[1].subs(t=t).simplify_full()) for t in range(-1,5)]
[(1, 0), (3, 2), (17, 12), (99, 70), (577, 408), (3363, 2378)]

TESTS:

sage: solve_diophantine(x^2 - y, x, y)
Traceback (most recent call last):
...
AttributeError: Please use a tuple or list for several variables.

sqrt(hold=False)

Return the square root of this expression

EXAMPLES:

sage: var('x, y')
(x, y)
sage: SR(2).sqrt()
sqrt(2)
sage: (x^2+y^2).sqrt()
sqrt(x^2 + y^2)
sage: (x^2).sqrt()
sqrt(x^2)

Using the hold parameter it is possible to prevent automatic evaluation:

sage: SR(4).sqrt()
2
sage: SR(4).sqrt(hold=True)
sqrt(4)

To then evaluate again, we currently must use Maxima via simplify():

sage: a = SR(4).sqrt(hold=True); a.simplify()
2

To use this parameter in functional notation, you must coerce to the symbolic ring:

sage: sqrt(SR(4),hold=True)
sqrt(4)
sage: sqrt(4,hold=True)
Traceback (most recent call last):
...
TypeError: _do_sqrt() got an unexpected keyword argument 'hold'

step(hold=False)

Return the value of the Heaviside step function, which is 0 for negative x, 1/2 for 0, and 1 for positive x.

EXAMPLES:

sage: x = var('x')
sage: SR(1.5).step()
1
sage: SR(0).step()
1/2
sage: SR(-1/2).step()
0
sage: SR(float(-1)).step()
0

Using the hold parameter it is possible to prevent automatic evaluation:

sage: SR(2).step()
1
sage: SR(2).step(hold=True)
step(2)

subs(*args, **kwds)

Substitute the given subexpressions in this expression.

EXAMPLES:

sage: var('x,y,z,a,b,c,d,f,g')
(x, y, z, a, b, c, d, f, g)
sage: w0 = SR.wild(0); w1 = SR.wild(1)
sage: t = a^2 + b^2 + (x+y)^3

Substitute with keyword arguments (works only with symbols):

sage: t.subs(a=c)
(x + y)^3 + b^2 + c^2
sage: t.subs(b=19, x=z)
(y + z)^3 + a^2 + 361

Substitute with a dictionary argument:

sage: t.subs({a^2: c})
(x + y)^3 + b^2 + c

sage: t.subs({w0^2: w0^3})
a^3 + b^3 + (x + y)^3

Substitute with one or more relational expressions:

sage: t.subs(w0^2 == w0^3)
a^3 + b^3 + (x + y)^3

sage: t.subs(w0 == w0^2)
(x^2 + y^2)^18 + a^16 + b^16

sage: t.subs(a == b, b == c)
(x + y)^3 + b^2 + c^2

Any number of arguments is accepted:

sage: t.subs(a=b, b=c)
(x + y)^3 + b^2 + c^2

sage: t.subs({a:b}, b=c)
(x + y)^3 + b^2 + c^2

sage: t.subs([x == 3, y == 2], a == 2, {b:3})
138

It can even accept lists of lists:

sage: eqn1 = (a*x + b*y == 0)
sage: eqn2 = (1 + y == 0)
sage: soln = solve([eqn1, eqn2], [x, y])
sage: soln
[[x == b/a, y == -1]]
sage: f = x + y
sage: f.subs(soln)
b/a - 1

Duplicate assignments will throw an error:

sage: t.subs({a:b}, a=c)
Traceback (most recent call last):
...
ValueError: duplicate substitution for a, got values b and c

sage: t.subs([x == 1], a = 1, b = 2, x = 2)
Traceback (most recent call last):
...
ValueError: duplicate substitution for x, got values 1 and 2

All substitutions are performed at the same time:

sage: t.subs({a:b, b:c})
(x + y)^3 + b^2 + c^2

Substitutions are done term by term, in other words Sage is not able to identify partial sums in a substitution (see trac ticket #18396):

sage: f = x + x^2 + x^4
sage: f.subs(x = y)
y^4 + y^2 + y
sage: f.subs(x^2 == y)             # one term is fine
x^4 + x + y
sage: f.subs(x + x^2 == y)         # partial sum does not work
x^4 + x^2 + x
sage: f.subs(x + x^2 + x^4 == y)   # whole sum is fine
y

Note that it is the very same behavior as in Maxima:

sage: E = 'x^4 + x^2 + x'
sage: subs = [('x','y'), ('x^2','y'), ('x^2+x','y'), ('x^4+x^2+x','y')]

sage: cmd = '{}, {}={}'
sage: for s1,s2 in subs:
....:     maxima.eval(cmd.format(E, s1, s2))
'y^4+y^2+y'
'y+x^4+x'
'x^4+x^2+x'
'y'

Or as in Maple:

sage: cmd = 'subs({}={}, {})'              # optional - maple
sage: for s1,s2 in subs:                   # optional - maple
....:     maple.eval(cmd.format(s1,s2, E)) # optional - maple
'y^4+y^2+y'
'x^4+x+y'
'x^4+x^2+x'
'y'

But Mathematica does something different on the third example:

sage: cmd = '{} /. {} -> {}'                    # optional - mathematica
sage: for s1,s2 in subs:                        # optional - mathematica
....:     mathematica.eval(cmd.format(E,s1,s2)) # optional - mathematica
'y^4+y^2+y'
'x^4+y+x'
'x^4+y'
'y'

TESTS:

No arguments return the same expression:

sage: t = a^2 + b^2 + (x+y)^3
sage: t.subs()
(x + y)^3 + a^2 + b^2

Similarly for a empty dictionary, empty tuples and empty lists:

sage: t.subs({}, (), [], ())
(x + y)^3 + a^2 + b^2

Invalid argument returns error:

sage: t.subs(5)
Traceback (most recent call last):
...
TypeError: not able to determine a substitution from 5

Substitutions with infinity:

sage: (x/y).subs(y=oo)
0
sage: (x/y).subs(x=oo)
Traceback (most recent call last):
...
RuntimeError: indeterminate expression: infinity * f(x) encountered.
sage: (x*y).subs(x=oo)
Traceback (most recent call last):
...
RuntimeError: indeterminate expression: infinity * f(x) encountered.
sage: (x^y).subs(x=oo)
Traceback (most recent call last):
...
ValueError: power::eval(): pow(Infinity, f(x)) is not defined.
sage: (x^y).subs(y=oo)
Traceback (most recent call last):
...
ValueError: power::eval(): pow(f(x), infinity) is not defined.
sage: (x+y).subs(x=oo)
+Infinity
sage: (x-y).subs(y=oo)
-Infinity
sage: gamma(x).subs(x=-1)
Infinity
sage: 1/gamma(x).subs(x=-1)
0

Verify that this operation does not modify the passed dictionary (trac ticket #6622):

sage: var('v t')
(v, t)
sage: f = v*t
sage: D = {v: 2}
sage: f(D, t=3)
6
sage: D
{v: 2}

Check if trac ticket #9891 is fixed:

sage: exp(x).subs(x=log(x))
x

Check if trac ticket #13587 is fixed:

sage: t = tan(x)^2 - tan(x)
sage: t.subs(x=pi/2)
Infinity
sage: u = gamma(x) - gamma(x-1)
sage: u.subs(x=-1)
Infinity

Check that the deprecated method subs_expr works as expected (see trac ticket #12834):

sage: var('x,y,z'); f = x^3 + y^2 + z
(x, y, z)
sage: f.subs_expr(x^3 == y^2, z == 1)
doctest:...: DeprecationWarning: subs_expr is deprecated. Please use
substitute instead.
See http://trac.sagemath.org/12834 for details.
2*y^2 + 1
sage: f.subs_expr({x^3:y^2, z:1})
2*y^2 + 1
sage: f = x^2 + x^4
sage: f.subs_expr(x^2 == x)
x^4 + x
sage: f = cos(x^2) + sin(x^2)
sage: f.subs_expr(x^2 == x)
cos(x) + sin(x)
sage: f(x,y,t) = cos(x) + sin(y) + x^2 + y^2 + t
sage: f.subs_expr(y^2 == t)
(x, y, t) |--> x^2 + 2*t + cos(x) + sin(y)
sage: f.subs_expr(x^2 + y^2 == t)
(x, y, t) |--> x^2 + y^2 + t + cos(x) + sin(y)

subs_expr(*args, **kwds)

Deprecated: Use substitute() instead. See trac ticket #12834 for details.

substitute(*args, **kwds)

Substitute the given subexpressions in this expression.

EXAMPLES:

sage: var('x,y,z,a,b,c,d,f,g')
(x, y, z, a, b, c, d, f, g)
sage: w0 = SR.wild(0); w1 = SR.wild(1)
sage: t = a^2 + b^2 + (x+y)^3

Substitute with keyword arguments (works only with symbols):

sage: t.subs(a=c)
(x + y)^3 + b^2 + c^2
sage: t.subs(b=19, x=z)
(y + z)^3 + a^2 + 361

Substitute with a dictionary argument:

sage: t.subs({a^2: c})
(x + y)^3 + b^2 + c

sage: t.subs({w0^2: w0^3})
a^3 + b^3 + (x + y)^3

Substitute with one or more relational expressions:

sage: t.subs(w0^2 == w0^3)
a^3 + b^3 + (x + y)^3

sage: t.subs(w0 == w0^2)
(x^2 + y^2)^18 + a^16 + b^16

sage: t.subs(a == b, b == c)
(x + y)^3 + b^2 + c^2

Any number of arguments is accepted:

sage: t.subs(a=b, b=c)
(x + y)^3 + b^2 + c^2

sage: t.subs({a:b}, b=c)
(x + y)^3 + b^2 + c^2

sage: t.subs([x == 3, y == 2], a == 2, {b:3})
138

It can even accept lists of lists:

sage: eqn1 = (a*x + b*y == 0)
sage: eqn2 = (1 + y == 0)
sage: soln = solve([eqn1, eqn2], [x, y])
sage: soln
[[x == b/a, y == -1]]
sage: f = x + y
sage: f.subs(soln)
b/a - 1

Duplicate assignments will throw an error:

sage: t.subs({a:b}, a=c)
Traceback (most recent call last):
...
ValueError: duplicate substitution for a, got values b and c

sage: t.subs([x == 1], a = 1, b = 2, x = 2)
Traceback (most recent call last):
...
ValueError: duplicate substitution for x, got values 1 and 2

All substitutions are performed at the same time:

sage: t.subs({a:b, b:c})
(x + y)^3 + b^2 + c^2

Substitutions are done term by term, in other words Sage is not able to identify partial sums in a substitution (see trac ticket #18396):

sage: f = x + x^2 + x^4
sage: f.subs(x = y)
y^4 + y^2 + y
sage: f.subs(x^2 == y)             # one term is fine
x^4 + x + y
sage: f.subs(x + x^2 == y)         # partial sum does not work
x^4 + x^2 + x
sage: f.subs(x + x^2 + x^4 == y)   # whole sum is fine
y

Note that it is the very same behavior as in Maxima:

sage: E = 'x^4 + x^2 + x'
sage: subs = [('x','y'), ('x^2','y'), ('x^2+x','y'), ('x^4+x^2+x','y')]

sage: cmd = '{}, {}={}'
sage: for s1,s2 in subs:
....:     maxima.eval(cmd.format(E, s1, s2))
'y^4+y^2+y'
'y+x^4+x'
'x^4+x^2+x'
'y'

Or as in Maple:

sage: cmd = 'subs({}={}, {})'              # optional - maple
sage: for s1,s2 in subs:                   # optional - maple
....:     maple.eval(cmd.format(s1,s2, E)) # optional - maple
'y^4+y^2+y'
'x^4+x+y'
'x^4+x^2+x'
'y'

But Mathematica does something different on the third example:

sage: cmd = '{} /. {} -> {}'                    # optional - mathematica
sage: for s1,s2 in subs:                        # optional - mathematica
....:     mathematica.eval(cmd.format(E,s1,s2)) # optional - mathematica
'y^4+y^2+y'
'x^4+y+x'
'x^4+y'
'y'

TESTS:

No arguments return the same expression:

sage: t = a^2 + b^2 + (x+y)^3
sage: t.subs()
(x + y)^3 + a^2 + b^2

Similarly for a empty dictionary, empty tuples and empty lists:

sage: t.subs({}, (), [], ())
(x + y)^3 + a^2 + b^2

Invalid argument returns error:

sage: t.subs(5)
Traceback (most recent call last):
...
TypeError: not able to determine a substitution from 5

Substitutions with infinity:

sage: (x/y).subs(y=oo)
0
sage: (x/y).subs(x=oo)
Traceback (most recent call last):
...
RuntimeError: indeterminate expression: infinity * f(x) encountered.
sage: (x*y).subs(x=oo)
Traceback (most recent call last):
...
RuntimeError: indeterminate expression: infinity * f(x) encountered.
sage: (x^y).subs(x=oo)
Traceback (most recent call last):
...
ValueError: power::eval(): pow(Infinity, f(x)) is not defined.
sage: (x^y).subs(y=oo)
Traceback (most recent call last):
...
ValueError: power::eval(): pow(f(x), infinity) is not defined.
sage: (x+y).subs(x=oo)
+Infinity
sage: (x-y).subs(y=oo)
-Infinity
sage: gamma(x).subs(x=-1)
Infinity
sage: 1/gamma(x).subs(x=-1)
0

Verify that this operation does not modify the passed dictionary (trac ticket #6622):

sage: var('v t')
(v, t)
sage: f = v*t
sage: D = {v: 2}
sage: f(D, t=3)
6
sage: D
{v: 2}

Check if trac ticket #9891 is fixed:

sage: exp(x).subs(x=log(x))
x

Check if trac ticket #13587 is fixed:

sage: t = tan(x)^2 - tan(x)
sage: t.subs(x=pi/2)
Infinity
sage: u = gamma(x) - gamma(x-1)
sage: u.subs(x=-1)
Infinity

Check that the deprecated method subs_expr works as expected (see trac ticket #12834):

sage: var('x,y,z'); f = x^3 + y^2 + z
(x, y, z)
sage: f.subs_expr(x^3 == y^2, z == 1)
doctest:...: DeprecationWarning: subs_expr is deprecated. Please use
substitute instead.
See http://trac.sagemath.org/12834 for details.
2*y^2 + 1
sage: f.subs_expr({x^3:y^2, z:1})
2*y^2 + 1
sage: f = x^2 + x^4
sage: f.subs_expr(x^2 == x)
x^4 + x
sage: f = cos(x^2) + sin(x^2)
sage: f.subs_expr(x^2 == x)
cos(x) + sin(x)
sage: f(x,y,t) = cos(x) + sin(y) + x^2 + y^2 + t
sage: f.subs_expr(y^2 == t)
(x, y, t) |--> x^2 + 2*t + cos(x) + sin(y)
sage: f.subs_expr(x^2 + y^2 == t)
(x, y, t) |--> x^2 + y^2 + t + cos(x) + sin(y)

substitute_expression(*args, **kwds)

Deprecated: Use substitute() instead. See trac ticket #12834 for details.

substitute_function(original, new)

Return this symbolic expressions all occurrences of the function original replaced with the function new.

EXAMPLES:

sage: x,y = var('x,y')
sage: foo = function('foo'); bar = function('bar')
sage: f = foo(x) + 1/foo(pi*y)
sage: f.substitute_function(foo, bar)
1/bar(pi*y) + bar(x)

TESTS:

Make sure trac ticket #17849 is fixed:

sage: ex = sin(x) + atan2(0,0,hold=True)
sage: ex.substitute_function(sin,cos)
arctan2(0, 0) + cos(x)
sage: ex = sin(x) + hypergeometric([1, 1], [2], -1)
sage: ex.substitute_function(sin,cos)
cos(x) + hypergeometric((1, 1), (2,), -1)

subtract_from_both_sides(x)

Return a relation obtained by subtracting x from both sides of this relation.

EXAMPLES:

sage: eqn = x*sin(x)*sqrt(3) + sqrt(2) > cos(sin(x))
sage: eqn.subtract_from_both_sides(sqrt(2))
sqrt(3)*x*sin(x) > -sqrt(2) + cos(sin(x))
sage: eqn.subtract_from_both_sides(cos(sin(x)))
sqrt(3)*x*sin(x) + sqrt(2) - cos(sin(x)) > 0

sum(*args, **kwds)

Return the symbolic sum \(\sum_{v = a}^b self\)

with respect to the variable \(v\) with endpoints \(a\) and \(b\).

INPUT:

• v - a variable or variable name

• a - lower endpoint of the sum

• b - upper endpoint of the sum

• algorithm - (default: 'maxima') one of

  • 'maxima' - use Maxima (the default)
  • 'maple' - (optional) use Maple
  • 'mathematica' - (optional) use Mathematica
  • 'giac' - (optional) use Giac

EXAMPLES:

sage: k, n = var('k,n')
sage: k.sum(k, 1, n).factor()
1/2*(n + 1)*n

sage: (1/k^4).sum(k, 1, oo)
1/90*pi^4

sage: (1/k^5).sum(k, 1, oo)
zeta(5)

A well known binomial identity:

sage: assume(n>=0)
sage: binomial(n,k).sum(k, 0, n)
2^n

And some truncations thereof:

sage: binomial(n,k).sum(k,1,n)
2^n - 1
sage: binomial(n,k).sum(k,2,n)
2^n - n - 1
sage: binomial(n,k).sum(k,0,n-1)
2^n - 1
sage: binomial(n,k).sum(k,1,n-1)
2^n - 2

The binomial theorem:

sage: x, y = var('x, y')
sage: (binomial(n,k) * x^k * y^(n-k)).sum(k, 0, n)
(x + y)^n

sage: (k * binomial(n, k)).sum(k, 1, n)
2^(n - 1)*n

sage: ((-1)^k*binomial(n,k)).sum(k, 0, n)
0

sage: (2^(-k)/(k*(k+1))).sum(k, 1, oo)
-log(2) + 1

Summing a hypergeometric term:

sage: (binomial(n, k) * factorial(k) / factorial(n+1+k)).sum(k, 0, n)
1/2*sqrt(pi)/factorial(n + 1/2)

We check a well known identity:

sage: bool((k^3).sum(k, 1, n) == k.sum(k, 1, n)^2)
True

A geometric sum:

sage: a, q = var('a, q')
sage: (a*q^k).sum(k, 0, n)
(a*q^(n + 1) - a)/(q - 1)

The geometric series:

sage: assume(abs(q) < 1)
sage: (a*q^k).sum(k, 0, oo)
-a/(q - 1)

A divergent geometric series. Do not forget to \(forget\) your assumptions:

sage: forget()
sage: assume(q > 1)
sage: (a*q^k).sum(k, 0, oo)
Traceback (most recent call last):
...
ValueError: Sum is divergent.

This summation only Mathematica can perform:

sage: (1/(1+k^2)).sum(k, -oo, oo, algorithm = 'mathematica')     # optional - mathematica
pi*coth(pi)

Use Giac to perform this summation:

sage: (sum(1/(1+k^2), k, -oo, oo, algorithm = 'giac')).factor()       # optional - giac
pi*(e^(2*pi) + 1)/((e^pi + 1)*(e^pi - 1))

Use Maple as a backend for summation:

sage: (binomial(n,k)*x^k).sum(k, 0, n, algorithm = 'maple')      # optional - maple
(x + 1)^n

Note

1. Sage can currently only understand a subset of the output of Maxima, Maple and Mathematica, so even if the chosen backend can perform the summation the result might not be convertable into a usable Sage expression.

TESTS:

Check that the sum in trac ticket #10682 is done right:

sage: sum(binomial(n,k)*k^2, k, 2, n)
1/4*(n^2 + n)*2^n - n

This sum used to give a wrong result (trac ticket #9635) but now gives correct results with all relevant assumptions:

sage: (n,k,j)=var('n,k,j')
sage: sum(binomial(n,k)*binomial(k-1,j)*(-1)**(k-1-j),k,j+1,n)
-sum((-1)^(-j + k)*binomial(k - 1, j)*binomial(n, k), k, j + 1, n)
sage: assume(j>-1)
sage: sum(binomial(n,k)*binomial(k-1,j)*(-1)**(k-1-j),k,j+1,n)
1
sage: forget()
sage: assume(n>=j)
sage: sum(binomial(n,k)*binomial(k-1,j)*(-1)**(k-1-j),k,j+1,n)
-sum((-1)^(-j + k)*binomial(k - 1, j)*binomial(n, k), k, j + 1, n)
sage: forget()
sage: assume(j==-1)
sage: sum(binomial(n,k)*binomial(k-1,j)*(-1)**(k-1-j),k,j+1,n)
1
sage: forget()
sage: assume(j<-1)
sage: sum(binomial(n,k)*binomial(k-1,j)*(-1)**(k-1-j),k,j+1,n)
-sum((-1)^(-j + k)*binomial(k - 1, j)*binomial(n, k), k, j + 1, n)
sage: forget()

Check that trac ticket #16176 is fixed:

sage: n = var('n')
sage: sum(log(1-1/n^2),n,2,oo)
-log(2)

tan(hold=False)

EXAMPLES:

sage: var('x, y')
(x, y)
sage: tan(x^2 + y^2)
tan(x^2 + y^2)
sage: tan(sage.symbolic.constants.pi/2)
Infinity
sage: tan(SR(1))
tan(1)
sage: tan(SR(RealField(150)(1)))
1.5574077246549022305069748074583601730872508

To prevent automatic evaluation use the hold argument:

sage: (pi/12).tan()
-sqrt(3) + 2
sage: (pi/12).tan(hold=True)
tan(1/12*pi)

This also works using functional notation:

sage: tan(pi/12,hold=True)
tan(1/12*pi)
sage: tan(pi/12)
-sqrt(3) + 2

To then evaluate again, we currently must use Maxima via simplify():

sage: a = (pi/12).tan(hold=True); a.simplify()
-sqrt(3) + 2

TESTS:

sage: SR(oo).tan()
Traceback (most recent call last):
...
RuntimeError: tan_eval(): tan(infinity) encountered
sage: SR(-oo).tan()
Traceback (most recent call last):
...
RuntimeError: tan_eval(): tan(infinity) encountered
sage: SR(unsigned_infinity).tan()
Traceback (most recent call last):
...
RuntimeError: tan_eval(): tan(infinity) encountered

tanh(hold=False)

Return tanh of self.

We have \(\tanh(x) = \sinh(x) / \cosh(x)\).

EXAMPLES:

sage: x.tanh()
tanh(x)
sage: SR(1).tanh()
tanh(1)
sage: SR(0).tanh()
0
sage: SR(1.0).tanh()
0.761594155955765
sage: maxima('tanh(1.0)')
0.7615941559557649
sage: plot(lambda x: SR(x).tanh(), -1, 1)
Graphics object consisting of 1 graphics primitive

To prevent automatic evaluation use the hold argument:

sage: arcsinh(x).tanh()
x/sqrt(x^2 + 1)
sage: arcsinh(x).tanh(hold=True)
tanh(arcsinh(x))

This also works using functional notation:

sage: tanh(arcsinh(x),hold=True)
tanh(arcsinh(x))
sage: tanh(arcsinh(x))
x/sqrt(x^2 + 1)

To then evaluate again, we currently must use Maxima via simplify():

sage: a = arcsinh(x).tanh(hold=True); a.simplify()
x/sqrt(x^2 + 1)

TESTS:

sage: SR(oo).tanh()
1
sage: SR(-oo).tanh()
-1
sage: SR(unsigned_infinity).tanh()
Traceback (most recent call last):
...
RuntimeError: tanh_eval(): tanh(unsigned_infinity) encountered

taylor(*args)

Expand this symbolic expression in a truncated Taylor or Laurent series in the variable \(v\) around the point \(a\), containing terms through \((x - a)^n\). Functions in more variables is also supported.

INPUT:

• *args - the following notation is supported
  • x, a, n - variable, point, degree
  • (x, a), (y, b), n - variables with points, degree of polynomial

EXAMPLES:

sage: var('a, x, z')
(a, x, z)
sage: taylor(a*log(z), z, 2, 3)
1/24*a*(z - 2)^3 - 1/8*a*(z - 2)^2 + 1/2*a*(z - 2) + a*log(2)

sage: taylor(sqrt (sin(x) + a*x + 1), x, 0, 3)
1/48*(3*a^3 + 9*a^2 + 9*a - 1)*x^3 - 1/8*(a^2 + 2*a + 1)*x^2 + 1/2*(a + 1)*x + 1

sage: taylor (sqrt (x + 1), x, 0, 5)
7/256*x^5 - 5/128*x^4 + 1/16*x^3 - 1/8*x^2 + 1/2*x + 1

sage: taylor (1/log (x + 1), x, 0, 3)
-19/720*x^3 + 1/24*x^2 - 1/12*x + 1/x + 1/2

sage: taylor (cos(x) - sec(x), x, 0, 5)
-1/6*x^4 - x^2

sage: taylor ((cos(x) - sec(x))^3, x, 0, 9)
-1/2*x^8 - x^6

sage: taylor (1/(cos(x) - sec(x))^3, x, 0, 5)
-15377/7983360*x^4 - 6767/604800*x^2 + 11/120/x^2 + 1/2/x^4 - 1/x^6 - 347/15120

TESTS:

Check that ticket trac ticket #7472 is fixed (Taylor polynomial in more variables):

sage: x,y=var('x y'); taylor(x*y^3,(x,1),(y,1),4)
(x - 1)*(y - 1)^3 + 3*(x - 1)*(y - 1)^2 + (y - 1)^3 + 3*(x - 1)*(y - 1) + 3*(y - 1)^2 + x + 3*y - 3
sage: expand(_)
x*y^3

test_relation(ntests=20, domain=None, proof=True)

Test this relation at several random values, attempting to find a contradiction. If this relation has no variables, it will also test this relation after casting into the domain.

Because the interval fields never return false positives, we can be assured that if True or False is returned (and proof is False) then the answer is correct.

INPUT:

• ntests – (default 20) the number of iterations to run
• domain – (optional) the domain from which to draw the random values defaults to CIF for equality testing and RIF for order testing
• proof – (default True) if False and the domain is an interval field, regard overlapping (potentially equal) intervals as equal, and return True if all tests succeeded.

OUTPUT:

Boolean or NotImplemented, meaning

• True – this relation holds in the domain and has no variables.
• False – a contradiction was found.
• NotImplemented – no contradiction found.

EXAMPLES:

sage: (3 < pi).test_relation()
True
sage: (0 >= pi).test_relation()
False
sage: (exp(pi) - pi).n()
19.9990999791895
sage: (exp(pi) - pi == 20).test_relation()
False
sage: (sin(x)^2 + cos(x)^2 == 1).test_relation()
NotImplemented
sage: (sin(x)^2 + cos(x)^2 == 1).test_relation(proof=False)
True
sage: (x == 1).test_relation()
False
sage: var('x,y')
(x, y)
sage: (x < y).test_relation()
False

TESTS:

sage: all_relations = [op for name, op in sorted(operator.__dict__.items()) if len(name) == 2]
sage: all_relations
[<built-in function eq>, <built-in function ge>, <built-in function gt>, <built-in function le>, <built-in function lt>, <built-in function ne>]
sage: [op(3, pi).test_relation() for op in all_relations]
[False, False, False, True, True, True]
sage: [op(pi, pi).test_relation() for op in all_relations]
[True, True, False, True, False, False]

sage: s = 'some_very_long_variable_name_which_will_definitely_collide_if_we_use_a_reasonable_length_bound_for_a_hash_that_respects_lexicographic_order'
sage: t1, t2 = var(','.join([s+'1',s+'2']))
sage: (t1 == t2).test_relation()
False
sage: (cot(pi + x) == 0).test_relation()
NotImplemented

Check that trac ticket #18896 is fixed:

sage: m=540579833922455191419978421211010409605356811833049025*sqrt(1/2)
sage: m1=382247666339265723780973363167714496025733124557617743
sage: (m==m1).test_relation(domain=QQbar)
False
sage: (m==m1).test_relation()
False

trailing_coeff(s)

Return the trailing coefficient of s in self, i.e., the coefficient of the smallest power of s in self.

EXAMPLES:

sage: var('x,y,a')
(x, y, a)
sage: f = 100 + a*x + x^3*sin(x*y) + x*y + x/y + 2*sin(x*y)/x; f
x^3*sin(x*y) + a*x + x*y + x/y + 2*sin(x*y)/x + 100
sage: f.trailing_coefficient(x)
2*sin(x*y)
sage: f.trailing_coefficient(y)
x
sage: f.trailing_coefficient(sin(x*y))
a*x + x*y + x/y + 100

trailing_coefficient(s)

Return the trailing coefficient of s in self, i.e., the coefficient of the smallest power of s in self.

EXAMPLES:

sage: var('x,y,a')
(x, y, a)
sage: f = 100 + a*x + x^3*sin(x*y) + x*y + x/y + 2*sin(x*y)/x; f
x^3*sin(x*y) + a*x + x*y + x/y + 2*sin(x*y)/x + 100
sage: f.trailing_coefficient(x)
2*sin(x*y)
sage: f.trailing_coefficient(y)
x
sage: f.trailing_coefficient(sin(x*y))
a*x + x*y + x/y + 100

trig_expand(full=False, half_angles=False, plus=True, times=True)

Expand trigonometric and hyperbolic functions of sums of angles and of multiple angles occurring in self. For best results, self should already be expanded.

INPUT:

• full - (default: False) To enhance user control of simplification, this function expands only one level at a time by default, expanding sums of angles or multiple angles. To obtain full expansion into sines and cosines immediately, set the optional parameter full to True.
• half_angles - (default: False) If True, causes half-angles to be simplified away.
• plus - (default: True) Controls the sum rule; expansion of sums (e.g. ‘sin(x + y)’) will take place only if plus is True.
• times - (default: True) Controls the product rule, expansion of products (e.g. sin(2*x)) will take place only if times is True.

OUTPUT:

A symbolic expression.

EXAMPLES:

sage: sin(5*x).expand_trig()
5*cos(x)^4*sin(x) - 10*cos(x)^2*sin(x)^3 + sin(x)^5
sage: cos(2*x + var('y')).expand_trig()
cos(2*x)*cos(y) - sin(2*x)*sin(y)

We illustrate various options to this function:

sage: f = sin(sin(3*cos(2*x))*x)
sage: f.expand_trig()
sin((3*cos(cos(2*x))^2*sin(cos(2*x)) - sin(cos(2*x))^3)*x)
sage: f.expand_trig(full=True)
sin((3*(cos(cos(x)^2)*cos(sin(x)^2) + sin(cos(x)^2)*sin(sin(x)^2))^2*(cos(sin(x)^2)*sin(cos(x)^2) - cos(cos(x)^2)*sin(sin(x)^2)) - (cos(sin(x)^2)*sin(cos(x)^2) - cos(cos(x)^2)*sin(sin(x)^2))^3)*x)
sage: sin(2*x).expand_trig(times=False)
sin(2*x)
sage: sin(2*x).expand_trig(times=True)
2*cos(x)*sin(x)
sage: sin(2 + x).expand_trig(plus=False)
sin(x + 2)
sage: sin(2 + x).expand_trig(plus=True)
cos(x)*sin(2) + cos(2)*sin(x)
sage: sin(x/2).expand_trig(half_angles=False)
sin(1/2*x)
sage: sin(x/2).expand_trig(half_angles=True)
(-1)^floor(1/2*x/pi)*sqrt(-1/2*cos(x) + 1/2)

ALIASES:

trig_expand() and expand_trig() are the same

trig_reduce(var=None)

Combine products and powers of trigonometric and hyperbolic sin’s and cos’s of x into those of multiples of x. It also tries to eliminate these functions when they occur in denominators.

INPUT:

• self - a symbolic expression
• var - (default: None) the variable which is used for these transformations. If not specified, all variables are used.

OUTPUT:

A symbolic expression.

EXAMPLES:

sage: y=var('y')
sage: f=sin(x)*cos(x)^3+sin(y)^2
sage: f.reduce_trig()
-1/2*cos(2*y) + 1/8*sin(4*x) + 1/4*sin(2*x) + 1/2

To reduce only the expressions involving x we use optional parameter:

sage: f.reduce_trig(x)
sin(y)^2 + 1/8*sin(4*x) + 1/4*sin(2*x)

ALIASES: trig_reduce() and reduce_trig() are the same

trig_simplify(expand=True)

Optionally expand and then employ identities such as \(\sin(x)^2 + \cos(x)^2 = 1\), \(\cosh(x)^2 - \sinh(x)^2 = 1\), \(\sin(x)\csc(x) = 1\), or \(\tanh(x)=\sinh(x)/\cosh(x)\) to simplify expressions containing tan, sec, etc., to sin, cos, sinh, cosh.

INPUT:

• self - symbolic expression
• expand - (default:True) if True, expands trigonometric and hyperbolic functions of sums of angles and of multiple angles occurring in self first. For best results, self should be expanded. See also expand_trig() to get more controls on this expansion.

ALIAS: trig_simplify() and simplify_trig() are the same

EXAMPLES:

sage: f = sin(x)^2 + cos(x)^2; f
cos(x)^2 + sin(x)^2
sage: f.simplify()
cos(x)^2 + sin(x)^2
sage: f.simplify_trig()
1
sage: h = sin(x)*csc(x)
sage: h.simplify_trig()
1
sage: k = tanh(x)*cosh(2*x)
sage: k.simplify_trig()
(2*sinh(x)^3 + sinh(x))/cosh(x)

In some cases we do not want to expand:

sage: f=tan(3*x)
sage: f.simplify_trig()
(4*cos(x)^2 - 1)*sin(x)/(4*cos(x)^3 - 3*cos(x))
sage: f.simplify_trig(False)
sin(3*x)/cos(3*x)

truncate()

Given a power series or expression, return the corresponding expression without the big oh.

INPUT:

• self – a series as output by the series() command.

OUTPUT:

A symbolic expression.

EXAMPLES:

sage: f = sin(x)/x^2
sage: f.truncate()
sin(x)/x^2
sage: f.series(x,7)
1*x^(-1) + (-1/6)*x + 1/120*x^3 + (-1/5040)*x^5 + Order(x^7)
sage: f.series(x,7).truncate()
-1/5040*x^5 + 1/120*x^3 - 1/6*x + 1/x
sage: f.series(x==1,3).truncate().expand()
-2*x^2*cos(1) + 5/2*x^2*sin(1) + 5*x*cos(1) - 7*x*sin(1) - 3*cos(1) + 11/2*sin(1)

unit(s)

Return the unit of this expression when considered as a polynomial in s.

See also content(), primitive_part(), and unit_content_primitive().

INPUT:

• s – a symbolic expression.

OUTPUT:

The unit part of a polynomial as a symbolic expression. It is defined as the sign of the leading coefficient.

EXAMPLES:

sage: (2*x+4).unit(x)
1
sage: (-2*x+1).unit(x)
-1
sage: (2*x+1/2).unit(x)
1
sage: var('y')
y
sage: (2*x - 4*sin(y)).unit(sin(y))
-1

unit_content_primitive(s)

Return the factorization into unit, content, and primitive part.

INPUT:

• s – a symbolic expression, usually a symbolic variable. The whole symbolic expression self will be considered as a univariate polynomial in s.

OUTPUT:

A triple (unit, content, primitive polynomial)` containing the unit, content, and primitive polynomial. Their product equals self.

EXAMPLES:

sage: var('x,y')
(x, y)
sage: ex = 9*x^3*y+3*y
sage: ex.unit_content_primitive(x)
(1, 3*y, 3*x^3 + 1)
sage: ex.unit_content_primitive(y)
(1, 9*x^3 + 3, y)

variables()

Return sorted tuple of variables that occur in this expression.

EXAMPLES:

sage: (x,y,z) = var('x,y,z')
sage: (x+y).variables()
(x, y)
sage: (2*x).variables()
(x,)
sage: (x^y).variables()
(x, y)
sage: sin(x+y^z).variables()
(x, y, z)

zeta(hold=False)

EXAMPLES:

sage: x, y = var('x, y')
sage: (x/y).zeta()
zeta(x/y)
sage: SR(2).zeta()
1/6*pi^2
sage: SR(3).zeta()
zeta(3)
sage: SR(CDF(0,1)).zeta()  # abs tol 1e-16
0.003300223685324103 - 0.4181554491413217*I
sage: CDF(0,1).zeta()  # abs tol 1e-16
0.003300223685324103 - 0.4181554491413217*I
sage: plot(lambda x: SR(x).zeta(), -10,10).show(ymin=-3,ymax=3)

To prevent automatic evaluation use the hold argument:

sage: SR(2).zeta(hold=True)
zeta(2)

This also works using functional notation:

sage: zeta(2,hold=True)
zeta(2)
sage: zeta(2)
1/6*pi^2

To then evaluate again, we currently must use Maxima via simplify():

sage: a = SR(2).zeta(hold=True); a.simplify()
1/6*pi^2

TESTS:

sage: t = SR(1).zeta(); t
Infinity

class sage.symbolic.expression.ExpressionIterator

Bases: object

next()

x.next() -> the next value, or raise StopIteration

sage.symbolic.expression.is_Expression(x)

Return True if x is a symbolic Expression.

EXAMPLES:

sage: from sage.symbolic.expression import is_Expression
sage: is_Expression(x)
True
sage: is_Expression(2)
False
sage: is_Expression(SR(2))
True

sage.symbolic.expression.is_SymbolicEquation(x)

Return True if x is a symbolic equation.

EXAMPLES:

The following two examples are symbolic equations:

sage: from sage.symbolic.expression import is_SymbolicEquation
sage: is_SymbolicEquation(sin(x) == x)
True
sage: is_SymbolicEquation(sin(x) < x)
True
sage: is_SymbolicEquation(x)
False

This is not, since 2==3 evaluates to the boolean False:

sage: is_SymbolicEquation(2 == 3)
False

However here since both 2 and 3 are coerced to be symbolic, we obtain a symbolic equation:

sage: is_SymbolicEquation(SR(2) == SR(3))
True

sage.symbolic.expression.solve_diophantine(f, *args, **kwds)

Solve a Diophantine equation.

The argument, if not given as symbolic equation, is set equal to zero. It can be given in any form that can be converted to symbolic. Please see Expression.solve_diophantine() for a detailed synopsis.

EXAMPLES:

sage: R.<a,b> = PolynomialRing(ZZ); R
Multivariate Polynomial Ring in a, b over Integer Ring
sage: solve_diophantine(a^2-3*b^2+1)
[]
sage: solve_diophantine(a^2-3*b^2+2)
(1/2*sqrt(3)*(sqrt(3) + 2)^t - 1/2*sqrt(3)*(-sqrt(3) + 2)^t + 1/2*(sqrt(3) + 2)^t + 1/2*(-sqrt(3) + 2)^t,
 1/6*sqrt(3)*(sqrt(3) + 2)^t - 1/6*sqrt(3)*(-sqrt(3) + 2)^t + 1/2*(sqrt(3) + 2)^t + 1/2*(-sqrt(3) + 2)^t)