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Posted By: pradha s Member Level: Gold Posted Date: 27 Jun 2008
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2007 M.Sc(Industrial mathematics) Question paper
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DISTANCE EDUCATION
M.Sc. (Industrial Mathematics) DEGREE EXAMINATION, DECEMBER 2007.
REAL ANALYSIS
Time : Three hours Maximum : 100 marks
Answer any FIVE questions.
All questions carry equal marks.
1. (a) Show that a set E is open if and only if its complement is closed.
(b) Show that compact subsets of metric spaces are closed.
(c) Show that every K-cell is compact.
2. (a) Show that every perfect set in is uncountable.
(b) Show that a subset E of is connected if and only if and implies
3. (a) State and prove the mean value theorem.
(b) Suppose f is a continuous mapping of into and f is differentiable in (a, b). Prove that there exists such that .
4. (a) Define Riemann-Stieltjes integration.
(b) If f is continuous on , prove that on .
(c) Assume increases monotonically and on . Let f be a bounded real function on . Prove that if and only if and
5. (a) State and prove Cauchy criterion for uniform convergence.
(b) Suppose uniformly on a set E in a metric space. Let x be a limit point of E and suppose that . Prove that converges, and .
6. (a) Suppose is a sequence of functions, differentiable on and converges for some point on . If converges uniformly on , then prove that converges uniformly on to a function f, and
.
(b) State and prove the Weierstrass approximation theorem.
7. (a) State and prove the inverse function theorem.
(b) State and prove the Gauss Divergence theorem.
8. (a) Prove that the outer measure of an interval is its length.
(b) State and prove monotone convergence theorem.
———————
DISTANCE EDUCATION
M.Sc. (I.M.) DEGREE EXAMINATION, DECEMBER 2007.
COMPLEX ANALYSIS
Time : Three hours Maximum : 100 marks
Answer any FIVE questions.
All questions carry equal marks.
1. (a) Verify Cauchy-Riemann’s equations for the function .
(b) State and prove Abel’s theorem.
2. (a) Show that an analytic function in a region whose derivative vanishes identically must reduce to a constant.
(b) Prove that a linear transformation carries circles into circles.
3. (a) Compute for the positive sense of the circle.
(b) State and prove the integral formula.
4. State and prove Morera’s theorem.
5. (a) State and prove Taylor’s theorem.
(b) Show that the functions and have essential singularities at .
6. State and prove the Cauchy’s theorem.
7. (a) State and prove the residue theorem.
(b) Compute .
8. State and prove the Laurent series development.
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