North Maharashtra University 2008 B.Sc Mathematics S.Y. - MTH – 221 Functions of a Complex Variable. - Question 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) The lim [3x + i( 2x - 4y )] is - - - - - -
z ? two + 3i
2) Does lim
z
z exist
z ? 0
3) elaborate the points of discontinuties of f(z) =
2 2
2 3
2 - +
-
z z
z
4) Write the real and imaginary parts of f(z) = z3 where z = x+iy.
5) obtain the limit , lim [x + i( 2x + y )]
z ? 1- i
1) Does Continuity at a point imply differentiability there at. Justify by an example.
2) describe an analytic function .
3) describe singular points of an analytic function f(z).
4) obtain the singular points for the function f(z) =
( z )( z )
z
1 1
2
+ two +
-
5) describe a Laplace’s Didifferential formula for F( x, y ) .
6) What is harmonic function ?
7) What do you mean by f(z) is differentiable at ?
8) Is the function u =
2
1 .log (x2 + y2 ) harmonic?
9) When do you say f(z) tends to a limit as z tends to 0 z ?
10) State Cauchy- Riemann equations.
11) State the necessary condition for the function f(x) to be abalytic.
12) Every differential function is continuous . actual or False.
II) Multiple option ques. :
1) If lim [x+i(2x+y)] = p+iq , then (p,q) = - - - - - .
z ? 1- i
(i) (1,1) (ii) ( -1,1) (iii) (1,-1) (iv) (-1,-1)
2) The function f(z) = x2 y2
xy
+
when z ? 0 and f(0) =0 is
(i) Continuous at z = 0 , (ii) Discontinuous at z =0
(iii) Not predictable , (iv) Constant
3)A Continous function is ifferential :
(i) actual ,(ii) False.
(iii)True & False, (v) actual or False
1) A function F( x, y ) satisfying Laplace formula is called
(i) Analytic (ii) Holomorphic
(iii) Harmonic, (iv)Non-hormonic
2) Afunction f(z) = ez is
(i) Analytic everywhere , (ii) Analytic nowhere
(iii) only differentiable, (iv) None
2
3) If f(z) = u – iv is analytic in the z-plane , then the C-R equations satisfied by it’s real and
imaginary parts are ,
(i) u x y = u ; y x u = -v (ii) x y y x u = -v ,u = v
7) An analytic function with constant modulus is
(a) Constant , (b) not constant , (c) analytic , (d) None of these.

Earning:   Approval pending.