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Unformatted text preview: MATH 6322, Complex Analysis Fall 2011, HW#9 (all about normal family) Due date: Wed., Nov. 30, 2011 Problem 1 . Page 204, #20, #24 on the textbook. The following consists a small project about the normal family. You are first asked to prove the Martys theorem. You are then asked to prove the so-called Zalcmans theorem . The final goal is to prove a generalization of Montels theorem (which corresponds to the little picard theorem). Note that the original Montels theorem we learnt corresponds to the Liouvilles theorem (why?). Recall the Montels theorem we have learnt: a family of holomporphic functions on a region of C is a normal family (in the sense that every sequence of functions in the family admits a subsequence that converges uniformly on compact sets to an analytic function) if the family is locally bounded, that is, for every compact subset K there is a constant C K such that | f ( z ) | C K for all z K and every f in the family. Next, you are asked to prove Martys theorem, which gives a necessary and sufficient condition for a family of holomrphic functions on region of C to be a normal family in the extended sense that every sequence of functions in the family admits either a subsequence that converges uniformly on compact sets to an analytic function or a subsequence that converges...
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This note was uploaded on 02/21/2012 for the course MATH 4377 taught by Professor Staff during the Summer '08 term at University of Houston.
- Summer '08