Answer Happy

We will use the following complex analysis result: Theorem 0.1. Let U be an open subset of C and let {{n}n>i be a sequence of holomorphic functions on U such that i converges uniformly on compact subsets of U. Then II (1 + in) converges uniformly on compact subsets of U. In particular, the limit is a holomorphic function. The n function is defined as (T) = e* II (1 - e2min), for 7 € H. 1 Observe that, by the result quoted above, is a holomorphic function on H. 22
4. Show that n(-1/7)= V-in(). Hint: from the previous exercise n(-1/T) = c/-irn() for some c E C; evaluate at an appro- priate to pin down c.