2011.pdf - Answer ALL questions from Section A All questions from Section B may be attempted but only marks obtained on the best two solutions from

2011.pdf - Answer ALL questions from Section A All...

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Answer ALL questions from Section A. All questions from Section B may be attempted, but only marks obtained on the best two solutions from Section B will count. The use of an electronic calculator is not permitted in this examination. The following notation is used throughout: for any a C and r > 0, D ( a, r ) = { z : | z - a | < r } , D ( a, r ) = { z : | z - a | ≤ r } , D 0 ( a, r ) = { z : 0 < | z - a | < r } . Section A 1. Let f be a function holomorphic on a punctured disk D 0 ( z 0 , r ), z 0 C , r > 0. Describe the three types of isolated singularities of a function f by explaining how they are related to the principal part of its Laurent expansion. 2. (a) Suppose that g ( z ) has a removable singularity at the point z 0 . What is the residue of g at z 0 ? Justify your answer. (b) What type of singularity does the function f ( z ) = sinh z z 4 have at z 0 = 0? Justify your answer. Find the residue of f at z 0 . 3. (a) Prove that the disk D ( a, r ), a C , r > 0, is an open set. (b) Let A and B be open subsets of the complex plane. Prove that A B is also open.
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