Improve documentation and user interface of Fourier series

[egourgoulhon]

As discussed in https://groups.google.com/forum/#!topic/sage-devel/b1pWen3lrKA, the documentation of Fourier series as implemented in piecewise-defined functions needs some improvement. This ticket provides it, as well as a simplification of the user interface: the half-period is now an optional argument; if not provided, it is inferred from the domain of the piecewise-defined function:

sage: f = piecewise([((0, 2*pi), cos(x))]) 
sage: f.fourier_series_cosine_coefficient(1)  # results in TypeError in Sage 8.0
1
sage: f.fourier_series_cosine_coefficient(1, pi)  # compatible with Sage 8.0
1

The ticket also corrects two bugs:

• computation of Fourier coefficients when the domain width does not coincide with the period:

sage: f = piecewise([((0, 4*pi), cos(x))])
sage: f.fourier_series_cosine_coefficient(1, pi)  # yields 2 in Sage 8.0
1

• despite what it claimed, the method fourier_series_partial_sum did not return S_N(x) but S_N-1(x) (this is because the summation was governed by srange(1, N) instead of srange(1, N+1)); we have now

sage: f = piecewise([((0, 2*pi), cos(x))])
sage: f.fourier_series_partial_sum(1, pi)  # yields 0 in Sage 8.0
cos(x)

CC: @wdjoyner

Component: documentation

Keywords: Fourier series, piecewise functions

Author: Eric Gourgoulhon

Branch/Commit: 8cb8a4d

Reviewer: Richard L Lozes

Issue created by migration from https://trac.sagemath.org/ticket/23672

[egourgoulhon]

Branch: public/23672-Fourier_series

[egourgoulhon]

Commit: de1bb70

[egourgoulhon]

New commits:

de1bb70

Improve documentation and user interface of Fourier series

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

8cb8a4d

Rewrite the Fourier series sections in Sage Constructions

[sagetrac-git]

Changed commit from de1bb70 to 8cb8a4d

[rll2021]

comment:3

Changes documented with explanatory texts and tests. Now behaves as one expects of a Fourier series.

[egourgoulhon]

comment:4

Thank you for the review!

[rll2021]

comment:5

Filled in "Reviewers" field.

[rll2021]

Reviewer: Richard L Lozes

[vbraun]

Changed branch from public/23672-Fourier_series to 8cb8a4d