Alagappa University 2007 M.Sc Computer Science COMPLEX ANALYSIS - Question Paper

DISTANCE EDUCATION
M.Sc. DEGREE EXAMINATION, DECEMBER 2007.
Mathematics
COMPLEX ANALYSIS
Time : 3 hours Maximum : 100 marks
ans any 5 ques..
All ques. carry equal marks.
1. (a) Derive Cauchy-Riemann equations.
(b) State and prove Abel’s limit theorem.
2. (a) Prove that a transformation carries circles into circles.
(b) Find the linear transformation which carries the circle into , the point –2 to the origin, and the origin to .
3. (a) State and prove Cauchy’s integral formula.
(b) State and prove Cauchy’s theorem for a rectangle.
4. (a) State and prove Maximum principle.
(b) State and prove Morera’s theorem.
5. (a) State and prove Cauchy’s residue theorem.
(b) State and prove arguments principal.
(c) Evaluate .
6. (a) State and prove Mittag-Leffler Theorem.
(b) Obtain the Jenson’s formula.
7. (a) Prove that a discrete module consists either of zero alone, of the integral multiples of a single complex number or of all linear combinations with integral coefficients of 2 numbers with normal ratio .
(b) Prove that any 2 bases of the identical module are connected by a unimodular transformation.
8. (a) Prove that .
(b) Derive the 1st order differential formula for .

Earning:   Approval pending.