Firstranker's choice
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Roll No. [ ] [T TTTTT] Total No. of Pages : 02
Total No. of Questions : 07
B.Sc. (Computer Science) (2013 & Onwards) (Sem.-6)
REAL ANALYSIS
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Subject Code : BCS-601
M.Code : 72781
Time : 3 Hrs. Max. Marks : 60
INSTRUCTIONS TO CANDIDATES :
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1. SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks each.
2. SECTION-B contains SIX questions carrying TEN marks each and students have to attempt any FOUR questions.
SECTION-A
1. Write briefly :
a) State Weierstrass M-test for uniform convergence of sequence of functions.
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b) Prove that f(z) = Z is not analytic anywhere.
c) Show that f'(z) = 1/z is not uniformly continuous in the region | z | < 1.
d) Show that cross ratio is invariant under bilinear transformation.
e) Determine the angle of rotation at z = (1 + i)/2 under the mapping w = z*.
f) Find the radius of convergence of the series Σzln! xn.
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g) Determine values of a, b such that z = ax2 + by2 is harmonic function.
h) Prove that Σan is uniformly convergent on [0,1] if Σan converges uniformly in [0, 1].
i) Discuss the convergence and uniform convergence of sequence {fn}.
j) Determine the linear functional transformation that maps z1 = 0, z2 = 0, z3 = 1 onto w1=—1, w2=—1, w3 =1, respectively.
1| M-72781 (53)-1978
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Firstranker's choice
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SECTION-B
2. State and prove Cauchy’s General Principle of uniform convergence.
3. Show that the sequence {fn}, where fn(x)= converges uniformly on R.
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1+ nx2
4. Examine the convergence of ?01 xnlogxdx
5. a) Prove that necessary condition for f (z) =u + iv, z = X + iy, to be analytic in a domain D are that ux = vy and uy = —vx.
b) If u = ex (x cosy—y siny), find the analytic function u + i v.
6. If f(z) is an analytic function of z in a region D of the plane and f'(z) ? 0 inside D, show that the mapping w = f'(z) is conformal at the points of D.
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7. Find Fourier expansion of f(x)=
x—p, —p<x<0
p—x, 0<x<p
NOTE : Disclosure of Identity by writing Mobile No. or Making of passing request on any page of Answer Sheet will lead to UMC against the Student.
2| M- 72781 (53)-1978
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This download link is referred from the post: PTU B-Sc CS-IT 2020 March Previous Question Papers