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 _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele