MTH305 : COMPLEX ANALYSIS II (2014) NATIONAL OPEN UNIVERSITY OF NIGERIA

14/16 AHMADU BELLO WAY, VICTORIA ISLAND, LAGOS

SCHOOL OF SCIENCE AND TECHNOLOGY

MARCH/APRIL 2014 EXAMINATION

COURSE CODE: MTH305

COURSE TITLE: COMPLEX ANALYSIS II

TIME ALLOWED: 3 HOURS

INSTRUCTION: ANSWER  ANY 4 QUESTIONS

1. (a) Evaluate each of the following using theorems on limits

5 marks

(ii)                                                       5 marks

(b)      Prove the                                                                   7 ½ marks

2. (a)   Prove that the function U = 2x(1- y) is harmonic                                          7 ½  marks

(b)   Find a function V such that f (z) = u + i v and express f (z) in terms of z       8  marks

3. (a)   Prove that  is uniformly continous in the region            7 ½ marks

(b) Using the definition,find the derivative of at the point where

(i)  (ii)                                                                                                10 marks

4.  (a) Expand is a Laurent series valid for (i) (ii)               7 ½ marks

(b) Find the value of the integral ,where  is the line segment from z = 0 to z = 2+i   10 marks

5. (a)  Expand  f (z) = Cos z in Taylor series about  and determine its region of convergence    7 ½  marks

(b) Find the value of the integral ,where  is the line segment from z = 0 to z = 2+i    10 marks

6.(a)   Expand  f (z) = Cos z in Taylor series about  and determine its region of convergence   7 ½  marks

(b)  For each of the following functions,determine the poles and residuces at the poles

(i)   (ii)                                                                                                                10 marks

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