Here is an example:
sage: f = lambda x:x^2
sage: f = Piecewise([[(-1,1),f]])
sage: x = var("x")
sage: n = var("n")
sage: f.fourier_series_cosine_coefficient(n,1)
((2*pi^2*n^2 - 4)*sin(pi*n) + 4*pi*n*cos(pi*n))/(pi^3*n^3)
sage: maxima.eval('declare(n,integer)')
'done'
sage: f.fourier_series_cosine_coefficient(n,1)
4*(-1)^n/(pi^2*n^2)
On Tue, Apr 15, 2008 at 11:58 AM, James Morrow
<[EMAIL PROTECTED]> wrote:
> Hello David,
>
> I am interested in the nth term of the Fourier series. In some cases there
> are simple formulas for that term (for example if the function is
> f(x)=x^2 on the interval [-pi,pi]). However I can't figure out how to tell
> sage to do this. I can ask it to compute int_{pi}^{pi} f(t)sin (n*t)dt)\pi,
> but it doesn't know that n is an integer and hence doesn't know that
> cos(pi*n) is (-1)^n or that sin(pi*n)=0. I can't figure out how to get it
> to simplify the answer.
>
> Jim
>
>
>
>
> On Tue, Apr 15, 2008 at 4:02 AM, David Joyner <[EMAIL PROTECTED]> wrote:
>
> > The module piecewise.py has fairly extensive functionality for the
> > computation of Fourier series of piecewise defined periodic functions.
> > It even allows filters. There are examples at
> > http://www.sagemath.org/doc/html/const/node12.html
> >
> http://www.sagemath.org/hg/sage-main/file/cc1e12a492fc/sage/functions/piecewise.py
> > I think it is well documented but if tere are any questions, please just
> ask.
> >
> > On Mon, Apr 14, 2008 at 7:47 PM, William Stein <[EMAIL PROTECTED]> wrote:
> >
> >
> >
> > >
> > > ---------- Forwarded message ----------
> > > From: James Morrow <[EMAIL PROTECTED]>
> > > Date: Mon, Apr 14, 2008 at 4:20 PM
> > > Subject: fourier series
> > > To: [EMAIL PROTECTED], William Stein <[EMAIL PROTECTED]>
> > >
> > >
> > > Hello,
> > >
> > > How do I use sage to compute a Fourier series? The general question
> > > is: How to compute the nth term? I guess I have to use maxima. How
> > > do I call maxima and then give it the imput? I don't just want (say)
> > > the 5th term, but the nth term. Do I make n a variable (integer
> > > variable)? Maybe I can't do it in this generality, but for instance
> > > this amounts to computing (int_{pi}^{pi} f(t)sin (n*t)dt)\pi, where I
> > > declare n is an integer variable.
> > >
> > > Jim
> > >
> > >
> > > --
> > > William Stein
> > > Associate Professor of Mathematics
> > > University of Washington
> > > http://wstein.org
> > >
> > > > > >
> > >
> >
>
>
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