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Complex Analysis – Final Exam June 16, 2009 A. Definitions: write down the definitions or the theorems and answer the questions. (10%) 1. State the definition of a residue. 2. What is line-circle-preserving property in linear fractional transformation? B. True or false (If it is false, explain briefly why it isn’t true). (15%) 1. The only possible singularities of a rational function are poles. 2. The function () ( )( ) 5/ s i n3 f zz =+ has a pole of order 4 at 0 z = . 3. If () ( ) ( ) 1/2 1/2 1/2 '1 1 fz z z z −− , then ( ) wf z = maps the upper half-plane 0 y > onto the interior of a rectangle. C. Find the Laurent series of 2 5 () (2 5 ) z fz z = −+ centered at 1 z = , in the following domains: (10%) 1. 11 z −< 2. 2 z < 3. 22 z −> D. Evaluate the following: (30%) 1. 1 1 z C dz zi e π ± , C: 4 z = 2. cot C zdz ± , C is the circle 32 z −= 3. 2 sinh C z dz z + ± , C is the rectangle defined by

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## This note was uploaded on 02/21/2012 for the course EE 101 taught by Professor 張捷力 during the Spring '07 term at National Taiwan University.

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97ä¸‹è¤‡è®ŠæœŸæœ«è€ƒ é›»&a

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