Tamil Nadu Open University (TNOU) 2009-3rd Year B.Sc Mathematics " REAL AND COMPLEX ANALYSIS " UG 470 BMS 07 - Question Paper
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UG-470 BMS-07B.Sc. DEGREE EXAMINATION -JANUARY 2009.
(AY - 2005-06 and 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. Prove that in any metric space, the union of any family of open sets is open.
2. Define a convergent sequence. Show that for a convergent sequence 4$n the limit is unique.
4. Show that any discrete metric space with more than one point is disconnected.
5. Show that the function /j= does not have a
z
limit as z - 0.
6. Find the bilinear transformation which maps the points z = -l, 1, oo respectively on w = -i, -1, i.
7. State and prove Liovilles theorem.
8. Define residue of / at an isolated singularity
Z + 1
calculate the residues of - at its poles.
z2-2z
PART B (5 x 10 = 50 marks)
Answer any FIVE questions.
Each question carries 10 marks.
9. State and prove Minkowskis inequality.
10. Show that 12 is complete.
11. Prove that f is continuous if and only if inverse image of every open set is open.
12. Show that any compact subset A of a metric space (M, d) is closed.
13. Show that the points zx and z2 are reflection points for the line az + az + f3 = 0 if and only if az1 + az2+ f3 = 0.
14. State and prove a sufficient condition for differentiability of complex valued function.
15. State and prove Taylors theorem.
16. State and prove Rouches theorem.
3 UG-470
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