MTH221 Functions of a Complex Variable B.Sc Question Bank : nmu.ac.in
Name of the University : North Maharashtra University
Degree : B.Sc
Department : Mathematics
Year : II
Name Of The Subject : MTH221 Functions of a Complex Variable.
Document type : Question Bank
Website : nmu.ac.in
Download Model/Sample Question Papers : https://www.pdfquestion.in/uploads/nmu.ac.in/5282-S.Y.B.Sc.%20Mathematics%20(%20MTH%20-%20221%20)%20Question%20Bank.pdf
Functions of a Complex Variable Model Paper :
North Maharashtra University ; Jalgaon.
Question Bank S.Y.B.Sc. Mathematics (Sem –II)
MTH – 221 . Functions of a Complex Variable.
Unit – 1 :
Functions of a Complex Variable.
Related : North Maharashtra University Abstract Algebra B.Sc Question Bank : www.pdfquestion.in/5276.html
I) Questions of Two marks :
1) The lim [3x + i( 2x – 4y )] is – z- 2 + 3i
2) Does lim z z exist z- 0
3) What are the points of discontinuties of f(z) = 2 2
4) Write the real and imaginary parts of f(z) = z3 where z = x+iy.
5) Find 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) Define an analytic function .
3) Define singular points of an analytic function f(z).
4) Find the singular points for the function f(z) = ( z )( z )
5) Define a Laplace’s Didifferential Equation 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 = 1 .log (x2 + y2 ) harmonic-
9) When do you say f(z) tends to a limit as z tends to 0 z –
II) Multiple Choice Questions :
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
3)A Continous function is ifferential :
(i) True ,(ii) False. (iii)True & False, (v) True or False
1) A function F( x, y ) satisfying Laplace equation 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
3) If f(z) = u – iv is analytic in the z-plane , then the C-R equations satisfied by
(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.
8) A Milne – Thomson method is used to construct
a) analytic function , b) Continuous function
c) differentiable function, d) None of these.
III) Questions for Four marks ;
1) Define he continuity of f(z) at z = 0 z and examine for continuity at z=0
1) Define limit of a function f(Z). Evaluate ; lim 4 16
2) Prove that a differentiable function is always continuous . Is the converse true –
3) Use the definition of limit to prove that, lim [x + i( 2x + y )] = 1+ i
5) Show that if lim f(z) exists, it is unique
6)Show that the function f(z) = z is continuous everywhere but not differentiable .
7) Define an analytic function . Give two examples of an analytic function.
8) Show that f(z) = 2 z is not analytic at any point in the z-plane .
9) State and prove the necessary condition for the f(z) to be analytic . Are these conditions sufficient ?
10) State and prove the sufficient conditions for the function f(z) to be analytic.
11) Prove that a necessary condition for a complex function w = f(z) = u(x,y)+iv(x,y) to be analytic at a point z =x+iy of its domain D is that at (x,y) the first order partial derivatives of u and v with respect to x and y exist and satisfy the Cauchy – Riemann equations : u x y = v and u v . y x Prove that for the function F(z) = U(x,y) + V(x, y), if the four partial derivatives Ux, Uy, Vx and Vy
I Questions of TWO marks :
1) Define Laplace Differential equation.
2) Define harmonic and conjugate harmonic functions.
3) True 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 equation.
9) State Cauchy-Goursat Theorem.
10) Define simple closed curve.
11) Define the term Simply connected region.
12) Define Jordan Curve.
13) State Jordan Curve theorem.
Questions of THREE marks :
1) If F(z) = u + iv is an analytic function then show that u and v both satisfy Laplace’s differential equation.
2) If F(z) = u(x,y) + iv(x, y) is an analytic function, show that F(z) is independent of ___ z .
3) Explain the Milne-Thomson’s method to construct an analytic function F(z) = u + iv when the real part u is given.
4) Explain the Milne-Thomson’s method to construct an analytic function F(z) = u + iv when the imaginary part v is given.
5) Find an analytic function F(z) = u + iv and express it in terms of z if u = x3 – 3xy2 + 3×2 – 3y2 + 1.
6) Find an analytic function F(z) = u + iv if, v = e-ysinx and F(0) = 1.