Madurai Kamraj University (MKU) 2006 M.Sc Mathematics REAL AND COMPLEX ANALYSIS - Question Paper
This Is for the MK DDU - "MSC Maths In in MKU", and please Refer to the Attached File,
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It's For "Msc in MATHS in MKU", It'a Course in "Madurai Kamaraj University" or %MK University%
The paper Name Is "REAL AND COMPLEX ANALYSIS"
6557/KA2 october 2007
Paper II REAL AND COMPLEX ANALYSIS
(For those who joined in July 2003 and after)
Time : Three hours Maximum : 100 marks
SECTION A (4 x 10 = 40 marks)
Answer any FOUR questions.
1. Prove that converges if p >1 and diverges if
np
p< 1.
2. If Ixtn is a series of complex numbers which converges absolutely then prove that every rearrangement of Ynn converges, and they all converge to the same sum.
3. State and prove Mean value theorem.
4. If f is continuous on [a, b] then prove that fe R(a) on {a, b].
6. Prove that the ratio (zlt z2,z3,z4) is real if and only if the four points he on a circle or on a straight line.
7. State and prove Weierstrass theorem.
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8. If f(z) is analytic and nonconstant in a region Q,
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then prove that its absolute value \f(z)\ has ho maximum in Q..
SECTION B (3 x 20 = 60 marks)
Answer any THREE questions.
9. State and prove
(a) Root test
(b) Ratio test. . * .
I . . ; > ;.= > i
oj
10. State and prove the fundamental theorem of calculus.
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12. (a) Prove that a mapping f of a metrif space X into a metric space Y is continuous if and onjf if /-1(C) is closed in X for every closed set C in Y.
(b) Suppose f is a continuous mapping of a compact metric space X into a metric space Y. Then prove that f(X) is compact.
13. State and prove Cauchy's theorem for rectangle.
14. Prove that the nonempty connected subsets of the real line are the intervals.
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