Bob Baillie asked
One of the first examples of Fourier series that a student encounters is something like this: f(x) = x/2, for -Pi < x < Pi.
The Fourier series is Sum[ (-1)^(n+1)/n * Sin[n x] ]. A second, more complicated example is: f(x) = -Pi/2, for -Pi < x < 0, f(x) = Pi/2, for 0 < x < Pi. This Fourier series is Sum[ (1 - (-1)^n)/n * Sin[n x] ]. Can one work backwards, from the coefficients to the function? About all you can do is replace sin(x) with Im(e^ix) and treat the problem as a hypergeometric series, of which you can only do the easiest cases. Rarely one might use a bibasic series with q=e^ix and p=1. To see the hopelessness of the general case, note that the Snowflake and more general fractals have fairly nice Fourier coefficients: http://gosper.org/fst.dvi, with (poorly scanned) figures http://gosper.org/fst1.png, http://gosper.org/fst1.png, http://gosper.org/fst1.png, and http://gosper.org/fst1.png . I have even seen a fairly simple Fourier series for the boundary of the Mandelbrot set. Just imagine writing a closed form for Im(Snowflake). That paper also gives the Fourier series for an arc repeated around the edges of a regular polygon or "star", and by letting the dimension parameter d=1, the Fourier series for the polygon itself. By studying these, we can learn to recognize the series for piecewise linear functions. --rwg PS, on the Lisp Machine I once had a demo where Macsyma computed in closed form the F.s. of a point alternately tracing an equilateral triangle and its inscribed circle, which I then animated as a sum of rotating vectors, using that machine's superior color graphics primitives. --------------------------------- Take the Internet to Go: Yahoo!Go puts the Internet in your pocket: mail, news, photos & more.