----- Original Message ---- From: "rwg@sdf.lonestar.org" <rwg@sdf.lonestar.org> To: math-fun <math-fun@mailman.xmission.com> Sent: Thursday, October 9, 2008 5:21:21 AM Subject: [math-fun] Sum z^2^n (Was: Theta_3(0,q) near the unit circle)

----- Original Message ----

From: "rwg@sdf.lonestar.org" <rwg@sdf.lonestar.org> To: math-fun <math-fun@mailman.xmission.com> Sent: Tuesday, October 7, 2008 4:02:59 AM Subject: [math-fun] Theta_3(0,q) near the unit circle

... PS, how can a natural boundary grow only like 1/sqrt(-log q)? The other thetas are *much* wilder. _______________________________________________

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.

Gene ?? Surely it blows up at z=+-1. Interestingly, it has an infinitude of roots (but trickily, no accumulation point) inside the unit circle: 0, -.65862675430016392241347283058, .120314841052762693451935272875 +- .934605942791339747870826388582 i, -.685206279747129651080553110888 +- .670534105899025904030180437365 i,... .

ParametricPlot3D[{r*Cos[t], r*Sin[t], Abs[Sum[(r*E^(I*t))^2^n, {n, 0, Infinity}]]}, {r, 0, 1}, {t, 0, 2*Pi}] reveals the first few and suggests a few more, but they're so abrupt Mma would need to rootfind to look convincing. If it found too many, the inner ones would be hidden behind a curtain of needles at the boundary, although in a super-accurate plot, those ought to be pretty transparent. --rwg _______________________________________________ Bill, did you overlook the n! in the denominator? Call the function g(z). Then g(1) = e, g'(1) = e^2, g''(1) = e^4 - e^2, etc.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. Gene