LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

XZ 22

SIXTH SEMESTER – APRIL 2008

MT 6600 – COMPLEX ANALYSIS

 

 

 

Date : 16/04/2008             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION – A

Answer ALL questions:                                                                    (10 x 2 = 20)

 

1. Prove that for any two complex numbers and ,

.

1. Verify Canchy – Riemann equations for the function

1. Show that is harmonic.
2. Define Möbius transformation.
3. State Cauchy – Goursat theorem.
4. State Cauchy’s Integral formula.
5. Evaluate where C is the circle  .
6. Find the Taylor’s series expansion about z=0.
7. Obtain the Laurent’s series for  in .
8. Find the residue of at .

 

SECTION – B

Answer any  FIVE questions:                                                          (5 x 8 = 40)

 

1. Prove that the function

 

is not differentiable at the origin, but Canchy-Riemann equations are satisfied there.

1. Show that the function is harmonic and find the corresponding analytic function.
2. Show that the transformation maps the circle onto a straight line .
3. Evaluate  if C is the positively oriented circle
4. State and prove Liouville’s theorem.
5. State and prove Residue theorem.
6. State and prove Taylor’s theorem.
7. Find the residues of  at its poles.

 

SECTION – C

Answer any  TWO questions:                                                           (2 x 20 = 40)

 

1. a) If  is a regular function of , prove that

1. b) Find the bilinear transformation that maps the points onto the points .

1. State and prove Cauchy’s theorem.
2. Expand in the regions.

1. i)

ii)

iii)

1. Using the method of Contour integration, prove that

i)

ii)

 

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