----- Original Message ---- From: Christian G. Bower <bowerc@usa.net> To: math-fun <math-fun@mailman.xmission.com> Sent: Saturday, October 11, 2008 3:24:43 PM Subject: Re: [math-fun] Sum z^2^n (Was: Theta_3(0,q) near the unit circle) ------ Original Message ------ Received: Fri, 10 Oct 2008 10:27:44 AM PDT From: Eugene Salamin <gene_salamin@yahoo.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Sum z^2^n (Was: Theta_3(0,q) near the unit circle)

Here's a nice challenge problem. Let f(z) = sum( z^(2^n) / n!,

n=0..infinity). We know that f(z) is analytic on the open unit disk and that f and all its derivatives are continuous and bounded on the closed unit disk. Find a function g(z) analytic on the open complement of the unit disk (possibly including infinity), with g and its derivatives continuous and bounded on the closed complement, and that agrees with f and its derivatives on the unit circle. The obvious extension method, Schwartz reflection, does not work, since f does not map the unit circle into itself.

Gene

Can't we just use f(1/z)? Christian ________________________________ Won't work. At z = exp(i t), 1/z = exp(- i t), so we have g(z) = f(1/z) = f(exp(- i t)) /= f(exp(i t)) = f(z). Gene