BSc Mathematics SEM VI ATKT 2018-19 ATKT Maths I Real & Complex Analysis 19 R Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q3 From Question 4, Attempt any THREE
2. Q4 Figures to the right indicate marks for the respective parts
3. Q1 Let {f,} be a sequence of Riemann integrable functions on [a, b]. If the series of functions converges uniformly to f on [a, b]. Show that f is Riemann integrable on [a, b] and Let be a sequence of real valued functions defined on a non-empty subset S of R. Show that converges uniformly to a function f if and only if for given > 04 a positive integer that | fin (x)| <6 State and prove Weierstrass M - test for uniform convergence of series of (12) ii By integrating a suitable power series over an interval [0,1], show that iii Discuss the uniform convergence of the sequence of functions {f,}on [0, 1], where [0,1] — Ris defined by f,(x) = x) iv Discuss the uniform convergence of the series of functions 8 marks
4. Q2 ai show that f(Z) = if and only if =0.Also using definition of differentiability, show that if exist then prove that the function F(z) = o(f (z)) has a derivative at and = Cis a domain in C. If u,v: Q- R are such that ssatisfy Cauchy Riemann equations and are continuous on Q, prove that f(z) = u(x, y) + iv (x, y) is analytic in Q b Using the definition, discuss differentiability of the function f where (12) analytic on a given domain D. If |f(z)| is constant on D, show that must be constant throughout D iii Show that f(z) = + iv(x,y) is analytic in a domain D if and only if v is a harmonic conjugate of u IV Find the image of the set |z| = 6, < under the reciprocal map w = in the extended complex plane 8 marks
5. Q3 be analytic everywhere inside and on a simple closed contour C, taken in the positive sense. If is any point interior to C, then prove that it Let C be a simple closed curve in the interior of the disc of convergence of the power series S(z) = — then prove that in the interior of the disk of convergence S'(z) = — Ifa function f is analytic at a given point then show that its derivatives of all (12) orders are analytic at that point too. Further suppose that a function f is analytic inside and on a positively oriented circle Cp, centered at and with radius R and if Mp denotes the maximum. value of |f(z)| on Cp then show that z, is a point inside the circle of convergence |z — Z)| = R of a power series — Zo)” then show that the series must be uniformly convergent in the closed disk |z — < Ry,where Ry = — Compute the residue of f(z) = at its simple poles State Theorem. Expand f(z) = a Laurent series for the 8 marks
6. Q4 a real power series has radius of convergence r, then show that it converges uniformly on [—s,s] < s <r ii Show that the sequence of functions converges uniformly on iii Test differentiability of the function f(z) = z|z| at (0,0) iv linear fractional transformation that maps 0, i, to —1, 0, 1 Evaluate f dz, where C is the circle |z —i| =2 Show that — dz| where |z 15 marks

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