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Tutorial Release 10.4 The Sage Development Team https://doc.sagemath.org/pdf/en/tutorial/sage_tutorial.pdf

Sage is free, open-source math software that supports research and teaching in algebra, geometry, number theory, cryptography, numerical computation, and related areas. Both the Sage development model and the technology in Sage itself are distinguished by an extremely strong emphasis on openness, community, cooperation, and collaboration: we are building the car, not reinventing the wheel. The overall goal of Sage is to create a viable, free, open-source alternative to Maple, Mathematica, Magma, and MATLAB.

This tutorial is the best way to become familiar with Sage in only a few hours. You can read it in HTML or PDF versions, or from the Sage notebook (click Help, then click Tutorial to interactively work through the tutorial from within Sage).

This work is licensed under a Creative Commons Attribution-Share Alike 3.0 License.

https://creativecommons.org/licenses/by-sa/3.0/

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ubuntu2204
Kernel: SageMath 10.4

2.4.5 Special functions

Several orthogonal polynomials and special functions are implemented, using both PARI and Maxima. These are documented in the appropriate sections (“Orthogonal polynomials” and “Special functions”, respectively) of the Sage reference manual.

x = polygen(QQ, 'x') chebyshev_U(2, x)
4*x^2 - 1
bessel_I(1, 1).n(250)
0.56515910399248502720769602760986330732889962162109200948029448947925564096
bessel_I(1, 1).n()
0.565159103992485
bessel_I(2, 1.1).n()
0.167089499251049

At this point, Sage has only wrapped these functions for numerical use. For symbolic use, please use the Maxima interface directly, as in the following example:

maxima.eval("f:bessel_y(v, w)")
'bessel_y(v,w)'
maxima.eval("diff(f,w)")
'(bessel_y(v-1,w)-bessel_y(v+1,w))/2'