Connection between Fourier transform and Taylor series

Both Fourier transform and Taylor series are means to represent functions in a different form.

What is the connection between these two? Is there a way to get from one to the other (and back again)? Is there an overall, connecting (geometric?) intuition?

Answer

Assume that the Taylor expansion f(x)=k=0akxk is convergent for some |x|>1. Then f can be extended in a natural way into the complex domain by writing f(z)=k=0akzk with z complex and |z|1. So we may look at f on the unit circle |z|=1. Consider f as a function of the polar angle ϕ there, i.e., look at the function F(ϕ):=f(eiϕ). This function F is 2π-periodic, and its Fourier expansion is nothing else but F(ϕ)=k=0akeikϕ where the ak are the Taylor coefficients of the “real” function xf(x) we started with.

Attribution
Source : Link , Question Author : vonjd , Answer Author : Christian Blatter

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