Tamil Nadu Open University (TNOU) 2008 B.Sc Mathematics 3rd year "Real And Complex Analysis" 319 - Question Paper
UG-319 BMS-07
B.Sc. DEGREE EXAMINATION - JUNE 2008. (AY 2005-2006, CY 2006 batches only)
Third Year Mathematics REAL AND COMPLEX ANALYSIS Time : 3 hours Maximum marks : 75
PART A (5 x 5 = 25 marks)
Answer any FIVE questions.
Each question carries 5 marks.
1. Show that any countably infinite set is equivalent to a proper subset of itself.
3. If f : R R and g : R R are both continuous functions on R and if h : R1 R2 is defined by h (x, j) = ( (x), g (y)), prove that h is continuous on
R2.
4. If (xn) is a cauchy sequence in a metric space M
and (xn) has a subsequence (x ) converging to x, then
show that (xn) converges to x .
5. If one of |a| and |b| is equal to 1, show that
a - b = 1 1 - ab
6. Prove that the function f (z) = ex (cos y - i sin y) is nowhere differentiable.
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where C is the semi circle |
z = 2eie, 0<0<n .
8. State and prove Cauchys inequality theorem.
Answer any FIVE questions.
Each question carries 10 marks.
9. State and prove Holders inequality.
10. Prove that any complete metric space is of second category.
11. Let (M1; d1) and (M2, d2) be two metric spaces. Then show that f : M1 M2 is continuous if and only if f (A)c f (A) for all A cMx.
12. Prove that a sub space of R is connected if and only if it is an interval.
13. Obtain Cauchy Riemann equations in polar coordinates.
14. Show that u = 2x - x3 + 3xy2 is harmonic and find its harmonic conjugate. Also find the corresponding analytic function.
15. State and prove Cauchys integral formula.
2n dfl
16. Evaluate [-, -1 < a < 1.
0 1 + a sin 9
3 UG-319
Define a complete metric space. Prove that any discrete metric space is complete.
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| Earning: Approval pending. |
