BSc Sem VI 2022 2023 Apr 2023 MATHEMATICS BASIC COMPLEX ANALYSIS Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q2 Use of a simple calculator is allowed
2. Q3 Figures to the right indicate marks
3. Q1 Attempt any one from the following. 8 marks
   • i) Let O C Cis in C. Rare such that then prove that f(z) = u(x, y) + iv (x,y) is analytic in 0 it) and Wg are points in z and w plans respectively then show that
   • (a) lim f(z) = if and only if lim — = 0
   • (b) lim f(z) = if and only if lim f = Wo
   • (c) lim f(z) = 00 if and only if = 0 B Attempt any two from the following. (12)
   • i) If a function f(z) = u(x, y) + is analytic in a domain D, then show that its component functions u and v are harmonic in D
   • ii) Use — 6 definition of limit to show that
   • iii) Let f be a analytic function throughout on a given domain D. If |f(z) is constant on D, show that f(z) must be constant on D Attempt any one from the following. (8)
4. Q1 State and prove extension of Cauchy’s Integral formula
   • ii) Suppose that a function f is analytic throughout a disk |z — < Ro, centered at Z) and with radius Ro. Then prove that f(z) has the power series representation f(z) = An(Z — , |Z — < Ro where An = i.e. the series converges to f(Z) when z lies in the stated B Attempt any two from the following. (12)
5. Q1 Let C denote a contour of length L and suppose that a function f(z) is piecewise continuous on C. If M is anon negative constant such that < M Vz C at which f(z) is defined then prove that
   • iii) Find a linear fractional transformation that maps the points 1, i, —1 onto the points —1,0,1 on the real axis Attempt any one from the following. (8) If aseries ay (Z — converges to f(z) at all points within the disc of convergence | < R then prove that it is the Taylor series expansion for f centered at Zo C bea simple closed curve in the interior of the disc of convergence of the power series S(z) = (Z — and let g(z) be any function which is continuous on C. Then prove that the series — can be integrated term by term over C and B Attempt any two from the following. (12)
   • i) If the power series An(Z — Zo)” converges for z = Zo), then prove that it is absolutely convergent for each z where R, = iit) radius of convergence R then find the radius of
   • iii) | Find Laurent series expansions in the domains: 1<|z|<
6. Q4 A Attempt any three questions from the following. 15 marks
   • i) Represent |z — = |z — as subsets of C in the plane where
   • ii) Show that z(t) = + tv and Re((z v) = 0 represents the same line in C
   • iii) all roots of the equation cos z = 2
   • iv) Determine whether the set of points 0, —4, —2i, —1 — 3i lies ona Find residue of f(z) at z = 0 where f(z) = (using the idea of vi) is bounded by x = 0,x

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