BSc Mathematics SEM VI 2016 17 2016-17 Real & Complex Analysis Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q3 From Question 4, Attempt any THREE gures to the right indicate marks for the respective > be sequence of differentiable real valued functions on [a, b] (8) cn that < > converges for some x9 and converges uniformly to function on [a,b]. Prove that converges uniformly on and uniform then f is differentiable on (a , b)-and f’ =9 on State and prove Weierstrass.M- test bi State and prove Cauchy’s criterion uniform convergence of the (12) sequence < fy, > of real series convergent in (0, 1]? Let 1] IR be pointwise limit of < Is f continuous < > converge uniformly f is a region R that is closed and (8) then show that there exists a non-negative integer M such that Also show that if exist , Let iv (%, exists at a point 2) = Xo + iyo prove -that the first order. partial derivatives of uand that:the not true. Also show that = (ux)z=z, + the definition, discuss differentiability of the function f(z) = 2? (12) analytic throughout on a given domain D. If |f(z)| is constant on show be constant on D component functions u and v are harmonic in image of the given set under the reciprocal map w = > in the State and prove Cauchy Integral 1 Suppose that a function f is analytic throug that f the 20 and with radius Ro. Then prove t function f is analytic at a given point of all orders are analytic at that point function f is analytic inside and oriented: centered at zy and with radius R Mg denotes of (z)| on Cg then show that li Prove that a power series (2 function S(z) at each point inside of convergence that any singular point’ of the Further determine the of each pole dnd find expansion in < Tal
2. Q4 i Does the < fi> fa(x) = uniformly 5) of the. function Construct a linear- fractional transformation that maps the points i, 00, 3 hon zero integer using a parameterisation of C

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