North Maharashtra University 2008 B.Sc Mathematics S.Y. - MTH – 221 Functions of a Complex Variable. - exam paper

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NORTH MAHARASHTRA UNIVERSITY, JALGAON

S.Y.B.Sc. Mathematics (Sem –II)
MTH – 221 . Functions of a Complex Variable.

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I ques. of 2 marks
1) describe Laplace Differential formula.
2) describe harmonic and conjugate harmonic functions.
3) actual or False:
i ) If F(z) is an analytic function of z, then F(z) depends on
___
z .
ii) If F(z) and
_____
F(z) are analytic functions of z, then F(z) is a constant.
iii) An analytic function with constant modulus is constant.
4) Is u = x2 - y2 a harmonic function? Justify.
5) Show that v(x, y) = x2 - y2 + x is harmonic function.
6) Show that u(x, y) = e-ysinx is a harmonic function.
7) Prove or disprove: u = y3 – 3x2y is a harmonic function.
8) Show that v = x3 - 3xy2 satisfies Laplace’s differential formula.
9) State Cauchy-Goursat Theorem.
10) describe simple closed curve.
11) describe the term Simply connected region.
12) describe Jordan Curve.
13) State Jordan Curve theorem.
14)
Evaluate ?
C
dz
z - a
1 where C is circle z - a = 2.
15)
Evaluate ?
+i
z dz
3
0
2 along the line x = 3y.
II Multiple option questions1 mark every
1) The harmonic conjugate of excosy is …………….
(a) excosy + c (b) exsiny + c (c) ex + c (d) None of these
2) The harmonic conjugate of e-ysinx is …………….
(a) e-ycosx + c (b) e-ysinx + c (c) e-x cosy (d) None of these
3)
The value of the integral ?
C
(12z2- 4/z) dz where C is the curve y = x3 - 3x2 + 4x – one joining
points (1,1) and (2,3) is provided by ……………….
(a) -156 + 58i (b) -156 - 58i (c) 50 (d) None of these
4)
The value of ?
1
0
z e2z dz will be
(a) e (b) one / four (e2 +1) (c) one / four (e2 -1) (d) None of these
5)
The value of the integral of one / z along a semicircular arc from -1 to one in the clockwise direction
will be ……………….
(a) zero (b) –pi (c) pi (d) None of these
ques. of 3 marks
5
1) If F(z) = u + iv is an analytic function then show that u and v both satisfy Laplace’s
differential formula.
2) If F(z) = u(x,y) + iv(x, y) is an analytic function, show that F(z) is independent of
___
z .
3) discuss the Milne-Thomson’s method to construct an analytic function F(z) = u + iv when the

Earning:   Approval pending.