Answer Happy

1. Here are some examples of analytic functions on the unit disc that cannot be extended analytically past the unit circle. The following definition is needed. Let

be a function defined in the unit disc D, with boundary circle C. A point w on C is said to be regular for S if there is an open neighborhood U of w and an analytic function g on U, so that I =g on DNU. A functionſ defined on D cannot be continued analytically past the unit circle if no point of C is regular for s. (a) Let f(x) =Σ 2" for 2 <1. TO Notice that the radius of convergence of the above series is 1. Show that f cannot be continued analytically past the unit disc. (Hint: Suppose 8 = 2mp/24, where p and k are positive integers. Let z = rel; then IS(re) as r - 1.1 - - (b) Fix 0 <O<0. Show that the analytic functionſ defined by f(x) =Σ 2 S(z) = 2*4,2" for [z] <1 1 . extends continuously to the unit circle, but cannot be analytically continued past the unit circle. (Hint: There is a nowhere differentiable function lurking in the background. See Chapter 4 in Book I.]