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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