Vinayaka Missions University 2008 M.Sc Mathematics REAL ANALYSIS (Correspondence ) - Question Paper

COURSE CODE – 2030507
PG DEGREE exam – SEP 2008
M.SC (MATHS)
REAL ANALYSIS
(For Candidate Admitted From Calendar Year 2007 Onwards)
Time: three hours Max. Marks: 75
part – A
ans all the Questions: 15 X one =15
1. Write down the formula for chain rule?
2. Write down the formula for Taylor’s?
3. What is inter mediate?
4. describe Riemann?
5. Write down Euler’s summation formula?
6. Write the necessary conditions for extensile?
7. describe stidjes integral?
8. Write order of integration?
9. Write down Riemann condition’s?
10. describe absolute convergence?
11. describe outer measure?
12. describe monotonic functions ?
13. describe absolutely continuity?
14. What is measure?
15. provide an example for leblegue measure?
Section–B
ans all questions: five X six =30
16. a) State and prove chain rule?
(Or)
b) State and prove Roller theorem?
17. a) State and prove Euler’s Summation formula?
(Or)
b) State and prove comparison theorem?
18. a) State and prove mean value theorem for Riemann – stieljes
integral?
(Or)
b) prove stieljes integral depending on a par meter?
19. a) State and prove Tannery’s theorem?
(Or)
b) describe i) Trigonometric functions.
ii) Hyperbolic function.
iii) Bernoulli’s numbers.
iv) measurable function.
20. a) describe i) Bounded measurable function.
ii) Finite measure function.
iii) Non-negative measurable function.
(Or)
b) discuss derivation of an integral?
Section-C
ans any 2 Questions: 2X15=30
21. State and prove Inter-mediate value theorem?
22. discuss necessary conditions for extensions of Riemann
stieljes?
23. State and prove fundamental theorem of integral calculus?
24. State and prove little woods 3 principles?
25. discuss general lebesgue integral?

Earning:   Approval pending.