----- Original Message ---- From: Dan Asimov <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Sent: Thursday, October 9, 2008 6:11:46 PM Subject: Re: [math-fun] Sum z^2^n (Was: Theta_3(0,q) near the unit circle) Gene wrote: << I wrote: << Is [Sum_n=1^oo z^(2^n)/n!] that function Hille shows converges both inside and outside the unit circle? And, IIRC, that at least radially is continuous at the points of convergence *on* the circle? (If not, there is such an example in that book, and it's very cool.)
I found the example you were looking for. The discussion begins on p. 21 of vol. 2.
Dumb of me to even wonder, since this weird example obviously can't be a power series in z. --Dan _____________________________________________________________________ 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