# Re: [sage-devel] Issues with Fourier series

```On Wed, Aug 9, 2017 at 7:03 AM, Eric Gourgoulhon <egourgoul...@gmail.com> wrote:
> Hi,
>
> It seems that the only way to compute with Sage the Fourier series of a
> periodic real function is through the methods fourier_series_* of
> piecewise-defined functions. Let us take a trivial example: the Fourier
> series of the cosine function:
>
> sage: f = piecewise([((0,2*pi), cos(x))])
>
> To get the cosine coefficient of order n of the Fourier series, one has to
> call the method fourier_series_cosine_coefficient with n and the half-period
> as argument:
>
> sage: f.fourier_series_cosine_coefficient(1, pi)
> 1
>
> So far so good. Now
>
> sage: f.fourier_series_partial_sum(1, pi)
> 0
>
> This is quite surprising at first glance and is actually the offset issue
> already discussed in #8603. The expect result is obtained via
>
> sage: f.fourier_series_partial_sum(2, pi)
> cos(x)
>
> To understand why one has to provide 2 as the first argument, let us have a
> look at the documentation:
>
> sage: f.fourier_series_partial_sum?
> Signature:      f.fourier_series_partial_sum(*args, **kwds)
> Docstring:
>    Returns the partial sum
>
>       f(x) sim frac{a_0}{2} + sum_{n=1}^N [a_ncos(frac{npi
>       x}{L}) + b_nsin(frac{npi x}{L})],
>
>    as a string.
>
> At the very least, this is very incomplete: there is no explicit relation
> between N and *args, nor between L and *args. One has to infer them from the
> provided examples. Moreover the specified output type is wrong: it is not a
> string but a symbolic expression:
>
> sage: type(f.fourier_series_partial_sum(2, pi))
> <type 'sage.symbolic.expression.Expression'>
>
> The documentation of fourier_series_cosine_coefficient is better because it
> has an INPUT section:
>
> sage: f.fourier_series_cosine_coefficient?
> Signature:      f.fourier_series_cosine_coefficient(*args, **kwds)
> Docstring:
>    Returns the n-th Fourier series coefficient of cos(npi x/L), a_n.
>
>    INPUT:
>
>    * "self" - the function f(x), defined over -L x L
>
>    * "n" - an integer n=0
>
>    * "L" - (the period)/2
>
>    OUTPUT: a_n = frac{1}{L}int_{-L}^L f(x)cos(npi x/L)dx
>
> However , it  needs some correction: n=0 looks like a typo here.
>
> Besides the documentation, another issue is the relation between the period
> and the piecewise function domain. Since one has to provide the half-period
> L as an argument to the fourier_series_* methods, this suggests that L is
> independent from the domain provided to piecewise. Let us then define
>
> sage: f = piecewise([((0,4*pi), cos(x))])
>
> This is the same cosine function as before but defined on a larger domain.
> Let us ask for the Fourier coefficient of order 1  by providing pi as the
> half-period, as above:
>
> sage: f.fourier_series_cosine_coefficient(1, pi)
> 2
>
> Boom!
> Having a look at the code of fourier_series_cosine_coefficient, one sees
> clearly why there is such an error:
>
> sage: f.fourier_series_cosine_coefficient??
>             from sage.all import cos, pi
>             x = SR.var('x')
>             result = 0
>             for domain, f in parameters:
>                 for interval in domain:
>                     a = interval.lower()
>                     b = interval.upper()
>                     result += (f*cos(pi*x*n/L)/L).integrate(x, a, b)
>             return SR(result).simplify_trig()
>
> The integral is taken over all the domain, while it should be taken over a
> single period only.
> A solution would be to deduce the half-period L from the domain (instead of
> getting it as an argument)  by writing something like
>
> L = (self.domain().sup() - self.domain().inf())/2
>
> I volunteer to write a ticket to improve the documentation and possibly the
> code. Any comment/suggestion prior to do so is welcome.
>```
```
Thanks for looking into this, Eric. I agree with your comments that the docs
(and possible the code) needs some improvement.

> Eric.
>
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