Get premium membership and access revision papers, questions with answers as well as video lessons.

Math 315: Complex Analysis I Question Paper

Math 315: Complex Analysis I

Course:Bachelor Of Science (General), Bachelor Of Education (Science & Arts)

Institution: Chuka University question papers

Exam Year:2013

?

CHUKA

UNIVERSITY

UNIVERSITY EXAMINATIONS
THIRD YEAR EXAMINATIONS FOR THE AWARD OF DEGREE OF
BACHELOR OF SCIENCE (GENERAL), BACHELOR OF EDUCATION (SCIENCE & ARTS)
MATH 315: COMPLEX ANALYSIS I
STREAMS: B.SC (GEN) BED (SCI&ARTS) Y3S2 TIME: 2 HOURS
DAY/DATE: MONDAY 22/4/2013 11.30 AM – 1.30 PM
INSTRUCTIONS:

Answer Question ONE and any other TWO

QUESTION 1 (30 MARKS)

1. (a) Show that for all ? ? C,??0,there exists ?^(-1) ? C such that ?.?^(-1)=Z^(-1) Z=1
[4 Marks]
(b) Define a complex conjugate ? ¯ of a complex number ? and show that
if ?_(1 ) and ?_2 are any two complex numbers,then ((Z_1/Z_2 ) ¯ )=? ¯_1/? ¯_2 [4 Marks]

(c) Find the expansion of the function f(z)=[(z-1)(z-2)]^(-1) in a Laurent’s
series that converges for |z|<1. [5 Marks]

(d) (i) State without proof De’ Moivre’s theorem.

(ii) Solve the equation leaving your answer in surd form: z^3-1=0
[5 Marks]

(e) Define the following terms:

(i) Closed and open path C
(ii) Connected region R [3 Marks]

(f) Find the value of tan??h^(-1) z.? [4 Marks]

(g) Show that the function U(x,y)=4xy-3x+2 is a harmonic. Construct the
corresponding analytic function. [5 Marks]

QUESTION 2 (20 MARKS)

(a) Derive an equation to show how one can get n-roots of unity of a complex number
?^n=1,where n ? ?. Hence or otherwise, show that the nth root of a complex number Z
is given by Z^(1/(n=)) r^(1/n) {cos((?+2px)/n)+i sin?((?+2px)/n) } where |Z|=r and ?=argZ,k=0,1,2,..,n-1 [10 Marks]

(b) Prove that if f(z) is analytic at a point c in the complex plane, then there is a power
series?_(n=0)^8¦?a_n (Z-c)^n ? where the coefficients are given by the formula
a_n=(f^n (c))/n! n=0,1,2,… and which converges to f(?) in every point in the
neighborhood of c, throughout which f is analytic, f(?)=?_(n=0)^8¦?f^n (c) ((f-c))/n!?
[10 Marks]

QUESTION 3 (20 MARKS)

(a) Define limit, continuity and derivative of a complex function f(z) at a given point in its
domain. [6 Marks]

(b) Prove that if the limit in (a) above exists, then it is unique. [6 Marks]

(c) Evaluate the following integrals:

(i) where c is the boundary of |z|=2 [3 Marks]

(ii) from z=0 to 4+2i along the curve Z=t^2+it. [5 Marks]

QUESTION 4 (20 MARKS)

(a) Define the following terms giving examples;

(i) an entire function
(ii) a singular point [4 Marks]

(b) (i) State and prove the residue theorem. [6 Marks]
(ii) Evaluate using the residue theorem ?¦(2z-1)/(z(z+1)(z-3)) dz where c is |z|=2.
[5 Marks]

(iii) Show that the singular points of the function f(z)=z+1/z^2 -2z are poles and
determine the order of each pole and the residue at that point. [5 Marks]

QUESTION 5 (20 MARKS)

(a) Derive the Cauchy – Riemann Equations in polar form. [10 Marks]

(b) Determine the circle of convergence of the series.
?_(n=1)^8¦(n!(z+pi)^n)/2^n [4 Marks]

(c) If f(t)=?(x,y)+if(x,y) represents a complex potential and ?=2x(1-y) determine the function f. [6 Marks]

---------------------------------------------------------------------------------------------------------------------

Math 315: Complex Analysis I Question Paper

More Question Papers

Popular Exams