Karnataka State Open University (KSOU) 2011-1st Sem M.Sc Mathematics (complex analysis) - Question Paper
Illlllllllllllllllllll Math 1.3
I Semester M.Sc. Examination, May 2011 MATHEMATICS Complex Analysis - I
Time : 3 Hours Max. Marks : 80
Instructions x Answer any fixe questions. Each question carry equal marks.
1. a) For each positive integer n, show that
2 n-l 1-Zn
1 + z + z +... + Z ---. (4+4+8)
1 - z
b) State and prove triangular inequality.
Z1 Z2
c) Prove that
1 - Zj z2
1 if either Izjl = 1 or |z2| = l. What exception must be
mode if zJ = z2 =1 ?
2. a) If Re z > 0, then prove that Re {zyjz2 - 1 j> 0. (8+8)
b) Show that zx and Zp corresponds to diametrically opposite points on the Riemann sphere if and only if ZjZ2 = - 1.
3. a) Find the general equation of a straight line. (6+4+6)
b) Show that if the equation z2 + az + p = 0 has a pair of conjugate complex roots, then a, P are both real and a 2 < 4p.
c) Define a continuity of a function. Prove that f(z) = u(x, y) + iv(x, y) on continuous at Zo = xo + iyo if and only if u and v are continuous at (xo, yo).
4. a) Deduce the polar form of a Cauchy-Riemann equations. (6+4+6)
. Where r > 0 and 0 <0 < 2% is
cos + ism
differentiable and find .
c) State and prove necessary and sufficient condition for a required to be convergent.
5. a) State and prove Weierstrass m-test. (6+6+4)
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b) Prove that the power series nan zn_1 obtained by differentially the power
n=0
CO
Z n
has same radius of convergence of as the original series
n=0
CO
.
n=0
c) Find the region of convergence of the series z i+-
n=0 V nS
oo A 1
n
6. a) Prove that bilinear transformation preserves cross ratio. (6+6+4)
b) Let f(z) be a function which is continuous on any continuous rectifiable curve C and W be any point of the complex place not lying in C. Then prove
that F(w) = Jdz is differentiable.
cz-w
f 2 * 2 o
c) Evaluate the integral J x - iy dz where C is the parabola y = 2x2 from (I, 2) to (2, 8).
7. a) State and prove Cauchy theorem for a disk. (8+8) b) State and prove Cauchy integral formula.
8. a) State and prove Liouvilles theorem and hence deduce fundamental theorem
of Algebra. (8+8)
b) State and prove Taylors theorem.
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