CMS: Smoothed Complex Series
----------------------------
CMS = "Complex Made Simple", the book by David C. Ullrich.

Consider the following function f defined by the complex power series:

  f(z) = Sum(n=0,oo) c_n z^n   ;  c_n and z complex, n = natural >= 0

Suppose this series has a (real) radius of convergence = r > 0.
Consider the function F defined by the slightly modified power series:

  F(z) = Sum(n=0,oo) C_n z^n  

Where the C_n are defined by:  C_n = s^(n^2).c_n  , for real 0 < s < 1

F(z) could be called a smoothed complex function (: take s close to 1)

Theorem (HdB). F(z) is an _entier_ function.

Proof(?). Define R = r/s^n . Then R approaches infinity for n -> oo
     and  |C_n.R^n| = s^(n^2).|c_n|.(r/s^n)^n = |c_n|.r^n = bounded.

Interpretation. Let s = exp(-(2.pi/N)^2/2) = exp(-(d.2.pi/L)^2/2) .
Then the series, when specified for z = exp(i.t) = the unit circle,
corresponds with a Fourier series expansion for an arbitrary Gaussian
smoothed closed curve with thickness d and length L, and it has almost
converged already for n = N. As argued in:

http://hdebruijn.soo.dto.tudelft.nl/jaar2004/Fransen3.pdf

Right or wrong? Any pointers to existing theory are quite welcome.

Han de Bruijn