Taylor expansion using the fft

[Fernando]

Hi there

Just want to post a simple Taylor expansion code based on Cauchy's integral
formula taking
 the contour to be a circle:

function coeff = taylor(N,r,f)

h = 2*pi/N;

n = 0:N-1; # index of coefficients        
th = n*h;  # step length around a circle

coeff = real(1./(N*(r.^n)).*fft(f(r*exp(i*th))));

octave:4> taylor(16,0.5,@(x) 1./(1-x))
ans =

 Columns 1 through 8:

   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000

 Columns 9 through 16:

   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000   1.0000


The problem is choosing the radius of your circle, to my knowledge theres no
optimal radius r.
Also you need to choose r such that the function f is analytic in that
region, in the example I chose r < 1 
as there is a pole at 1.

Just thought I share it since its so simple. Hope its useful.

Cheers 
Fernando

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