NATIONAL OPEN UNIVERSITY OF NIGERIA

Plot 91, Cadastral Zone, Nnamdi Azikiwe Express Way, Jabi-Abuja

FACULTY OF SCIENCES

January\February Examination 2018

Course Code: MTH305

Course Title: Complex Analysis II

Credit Unit: 3

Time Allowed: 3 Hours

Total Marks: 70%

INSTRUCTION: ANSWER QUESTION ONE(1) AND ANY FOUR (4)

QUESTIONS (TOTAL = 5 QUESTIONS IN ALL)

1(a) Expand f(z) = cos z by taylor series about the point (8marks)

1(b) Show that the following function are harmonic

(i) (3marks)

(ii) (3marks)

1(c) Given that

(3marks)

1(d) Express cos3 and Sin 3 in terms of sin and Cos only and state the real and imaginary parts. (5marks)

Total Marks = 22

(2a) Show that is an hyperbolic function (6marks)

(2b) Find the cube root of 8 in terms of complex number (6marks)

(3a) Verify that the real and imaginary parts of the function satisfy Cauchy-Riemann equation and deduce the analyticity of the function (8marks)

(3b) Prove that (4marks)

(4a) Solve: (6marks)

(4b) List six types of function (6marks)

(5a) Given that (i.e in polar form) Find the modulus of z and the principal argument of z.

(5b) Find also argz, if

(6a) Given that . Find a and b (6marks)

(6 b) If Find:

(3marks)

(3marks)

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