- #1

mikeph

- 1,235

- 18

Somewhat urgent question, I would normally try and do this myself but I have a feeling it will take a while, and I sort of need to be working through this pretty quickly, so any help much appreciated. Plus I might end up wasting half a day trying formulas on this.

I am expanding a function f(x + a sin(t)) as a Taylor series, so I get:

f(x + a sin (t)) = f(x) + (a sin t)f'(x) + (a sin t)^2/2! *f''(x) + ...

Now I am trying to decompose this into some sort of Fourier series (not exactly clear on final goal or method yet), so I need this in terms of:

f(x + a sin (t)) = [ ... ] + [ ... ]*sin(t) + [ ... ]*sin(2t) + ... + [ ... ]*cos(t) + [ ... ]*cos(2t) + ...

I am wondering if there is any simple way to do this. I have worked out the first one, eg:

sin^2(t) = 1/2 - 1/2 *cos(t), and was wondering if there are any simple recursion formulas or anything out there to help me find an expression for sin^n(t) in terms of single powers of sines and cosines of multiples of t?

Thanks for any help,

Mike