BSc Mathematics SEM VI 2018 19 May 2018-19 MATHEMATICS BASIC COMPLEX ANALYSIS Question Paper - Mumbai University | munotes

Questions asked in this paper

• 2. Figures to the right indicate full marks

1. Q1 Choose the correct alternative in each of the following: 20 marks
   • (a) 1 (b) i (c)-i (d) does not exist
   • ii. The image of a line under a fractional linear transformation is
   • (a) (b) a circle (c) a line or a circle (d) None of these
   • (a) continuous and bounded in |z| < 2
   • (b) continuous but not bounded in |z| < 2
   • (c) neither continuous nor bounded in 2
   • (d) continuous and bounded everywhere
   • (a) v and u are harmonic conjugates of each other
   • (b) wis a harmonic conjugate of v but v is not conjugate of u
   • (c) v is a harmonic conjugate of u but u is not a harmonic conjugate of v
   • (d) None of these
   • vi. thenz =
   • (c)i (d) none of these Vil. f é dz, where C is the circle |z| = 3, described in the positive sense is
   • (a) 2mi e? (b) 2mi (d) None of these Vill. Radius of convergence of the series iS
   • (b) 1 None of these
2. Q9 The poles of the function are at
   • (a) n is any integer (b) any integer
   • (c) nm, is any integer (d) none of these
3. Q10 The residue of f at z = 0 where f(z) =z cos= is
   • (a) (b) (c) (d) none of these any ONE question from the following: (08)
   • i. Let f(z) =u(x,y) + iv (x,y). If f (Z) exists point = x9 + iyo, then prove that the first order partial derivatives of u and v exist at Also show that = +
   • ii. If f (Zo), exist then prove that the function F(z) = o(f (z)) has a derivative at Z) and F'(z)) = If is differentiable at A, then show that f is. continuous at Let such that f is differentiable at Q, then show that da function such that f(z) = + f —
   • b) Attempt any TWO questions from the following: 12
   • i. Show that z(t) = z) + tv and Re((z = 0 represents the same
   • ii. If f'(z) = 0 everywhere on a domain D then show that f(z) must be
   • iii. Show that f(z) = z |z| is differentiable everywhere when f is treated as a function from R? R? but C differentiable only at z = 0
   • iv. If f(z) =8x-—x3 + i(x?y + y? — 8y) then determine points at which f is differentiable, f is analytic
4. Q3 a) Attempt any ONE question from the following: 8 marks
5. Q1 f analytic inside and ona simple, closed curve C, taken in the positive sense. Prove that f & f . Further state the result generalizing the formula to f"(z) Define complex sine and cosine functions. Also establish the following
   • b) Attempt any TWO questions from the following: Evaluate the integral f dz, where Find a Mobius transformation that maps i, 3 to 1/2 , —1,3 respectively 12
6. Q4 State Taylor’s theorem and also find Taylor series for f(z) = around
7. Q4 a) Attempt any ONE question from the following: 8 marks
8. Q1 If C is a simple closed curve in the interior of the disc of convergence of the power series S(z) = — and g(z) be any function which is continuous on C then prove that the series can be integrated term by term over C and
   • ii. If z, is a point inside the circle of convergence |z — = R of a power series — then show that the series must be uniformly convergent in the closed disk |z — < Ry, where = —
   • b) Attempt any TWO questions from the following: 12
9. Q1 Define the following terms: A removable singularity, A pole of order n, An essential singularity
   • ii. With the help of series, show _ that entire, use it to show that lim,_,9 lil. Laurent series representations of function f(z) = in the the real improper integral using the method of residue Attempt any FOUR questions from the following: (20)
   • a) Use Cauchy Riemann equations to check differentiability of f(z) = Re z
   • b) Show that w(x, y) is harmonic in some domain and find a harmonic conjugate Find image of. the set = <7} under the reciprocal map w = 1/z on the extended complex plane
   • d) Find values of z such that exp(2z — 1) = 1
   • e) Let has radius of convergence R. Find the radius of convergence of
   • f) Evaluate , where C is circle |z| = 1

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