Amrita Vishwa Vidyapeetham 2007 B.Sc Mathematics Complex Part Analysis - Question Paper

M.Sc. Dec 2007
Complex Part Analysis
Time : 3 hours Maximum : 100 marks
ans any 5 ques..
All ques. 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 decrease 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.

Earning:   Approval pending.