----- Original Message ---- From: Dan Asimov <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Sent: Thursday, October 9, 2008 11:40:46 AM Subject: Re: [math-fun] Sum z^2^n (Was: Theta_3(0,q) near the unit circle) Gene wrote: << Einar Hille gives the following example (Analytic Function Theory, vol. 1, p.133). sum( z^(2^n) / n!, n=0..infinity) This function and its derivatives of all orders are continuous and bounded in the closed unit disk. Yet the unit circle is a natural boundary. . . . . . . Hille shows that the n-th derivative grows with n so fast that a Taylor series about z=1 would have zero radius of convergence. The same applies at 2^n-th roots of unity, so the unit circle possesses a dense set of points that obstruct analytic continuation.
Is this 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.) --Dan ____________________________________________________________________ Dan, I found the example you were looking for. The discussion begins on p. 21 of vol. 2. Also notice the nice example in exercise 6 on that page, in which the pole in 1/z is spread out over a Cantor set to give a function continuous everywhere, analytic in the complement of the Cantor set, and its contour integral over a path enclosing the Cantor set is 2 pi i. Gene