BSc Mathematics SEM VI 2016 17 2016-17 Maths I Course 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 ai Let < f, > be sequence of differentiable real valued functions on [a, such that < > converges for some x9 (a,b) and < > converges uniformly to function gon [a,b]. Prove that <f, > converges uniformly on and f is uniform limit of </, > then f is differentiable on (a,b) and f’ = g on (a,b) ii State and prove Weierstrass M- test b State and prove Cauchy’s criterion for uniform convergence of the (12) sequence < > of functions of real numbers ii Examine whether = x"(1—2x")dx. Is the series x"(1 uniformly convergent in [0, 1]? Justify iv Let > IR be given by = x”. Let f be pointwise limit of < >. Is f continuous on [0,1]. Does < >. converge uniformly 8 marks
4. Q2 function f is continuous throughout a region R that is closed and bounded then show. that there exists a non-negative integer M such that that if exist , f(z) = + iv y). If exists at a point = x9 + iyo then prove that the first order partial derivatives of u and v exist at Show that the converse is not true. Also show that = + b i. Using the definition, discuss differentiability of the function f(z) = (12) 8 marks
   • ii. is analytic throughout on a given domain D. If |f(z)| is constant on D, show that f(z) must be constant on D If a function f(z) = + iv(x,y) is analytic in a domain D then show that its component functions u and v are harmonic in D Find the image of the given set under the reciprocal map w = in the
   • Q. P. Code: 05017
5. Q3 State and prove Cauchy Integral Theorem. ii Suppose that a function f is analytic throughout a disk < Ro, centered at and with radius Rp. Then prove that f(z) has the power series representation f(z) = — — where function f is analytic at a given point then show that its derivatives (12) of all 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 Z) and with radius R and if Mp denotes the maximum value of on Cp then show that < = ii Prove that a power series — Zo)” represents a continuous function S(z) at each point inside its circle of convergence |z — =r that any singular point of the function f(z) = pole Further determine the order m of each pole and find the corresponding State Laurent’s Theorem. For f(z) = , write Laurent series expansion in the 2 < |z 8 marks
6. Q4 the sequence < f,, f,(x) = converges uniformly For |x| < 1, show that = iii of the function f(z) =z Im z at (0,0) iv Construct a linear fractional transformation that maps the points i, 3 Evaluate dz where C is the circle =r, nisa non zero integer using a parameterisation of C Vl Evaluate J. dz where C: |z| = 2 15 marks

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