2007.pdf - UNIVERSITY COLLEGE LONDON EXAMINATION FOR INTERNAL STUDENTS MODULE CODE MATH2101 MODULE NAME Analysis 3 Complex Analysis DATE 08-May-07 TIME

2007.pdf - UNIVERSITY COLLEGE LONDON EXAMINATION FOR...

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UNIVERSITY COLLEGE LONDON EXAMINATION FOR INTERNAL STUDENTS MODULE CODE MATH2101 MODULE NAME Analysis 3: Complex Analysis DATE 08-May-07 TIME 10:00 TIME ALLOWED 2Hours 0Minutes 2006/07 -MATH21 01A-001-EXAM-196 ©2006 University College London TURN OVER
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All questions may be attempted but only marks obtained on the best four solutions will count. The use 01 an electronic calculator is not permitted in this examination. 1. (a) Let 1 be a complex valued function on a domain o c:c: Define each of the following: (i) 1 is holomorphic on D (ii) 1 has a root of multiplicity n at a point Zo E D. (b) Give an example of (i) a function 1 holomorphic everywhere except at ±1 (ii) an entire function 1 for which 1/1 fails to be holomorphic at precisely three points (iii) a function 1 which is not holomorphic anywhere. You do not need to justify your answers. (c) Let 1 be a holomorphic function of z = x + iy where x and yare real, and let 'U = Re i, v = Im f. State and prove the Cauchy - Riemann equations for f.
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