# Inverting a Taylor series

1. Apr 13, 2006

### eljose

of course if we know f(x) the function and is analytic near x=0 with this we could construct its Taylor series:

$$f(x)=a_{0}+a_{1}x+a_{2}x^{2}+...............$$

but the problem is..what happens if we know the $$a_{n}$$ but not f(x)?..well using Cauchy,s formula definition of the a we have:

$$a_{n}=\frac{n!}{2i\pi}\int_{C}dzf(z)z^{-(n+1)}$$

where z is a closed curve..if we choose C to be a circle or radius 1 the above formula becomes:

$$a_{n}=\frac{n!}{2i\pi}\int_{-\pi}^{\pi}dwf(e^{iw})e^{-iwn}$$

wich is nothing but a Fourier transform so we could invert it to obtain $$f(e^{iw})$$ and hence the function f(x)..but this is "correct"?..thanks.

Last edited: Apr 13, 2006