Taylor series

Taylor series express
a function $\bR \to \bC$ (or $\bC \to \bC$)
-infintesimally-
as $\sum a_n x^n$
...and requires (complex) smoothness at that point

Fourier series express
a function $S^1 \subset \bC \to \bC$
-globally-
as $\sum a_n z^n$
...and requires global integrability of stuff

I want to say:
- the Fourier coefficients of $\exp(f(ix))$
  -are- the Taylor coefficients of $f(x)$
Something like this is true, I think...
  just write out $f(x)=\sum a_n x^n$ and compute.

   %   %   %   %   %   %   %   %   %   %   %   %   %   %   %   %

I want to say that this a key aspect to understanding analysis
geometrically:
you can replace -infintesimal- analysis of $f$ at $x$
by -global- analysis of the tangent space at $x$
...and that's a key reason that global methods are legit.

See, ``in real life'' you're often working with a manifold or some
open subset of $\bR^n$, so you need to work locally.
...but if you concentrate at a point, you can look at the tangent
space, which is always all of $\bR^n$.
