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.