LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034.

M.Sc. DEGREE EXAMINATION – mathematics

SECOND SEMESTER – APRIL 2003

MT 2802 / M 823  –  COMPLEX ANALYSIS – II

 

24.04.2003

1.00 – 4.00                                                                                                      Max : 100 Marks

Answer ALL questions.

 

1. (a) Show that .                                                                              (8)

 

• Let Re Z_n >0 for all Prove  that  Converges to a non zero number if and only if the series                                                                             (8)

 

• (i) Obtain the Gauss formula for the Gamma function and show that
  éé

(ii)  If | z |  and p ³ o then prove that                            (7+2+8)

 

OR

• (i) Let f be a real valued function defined on  such that for all  x > 0.
  Suppose f(x) statistics the following properties.
• log f (x) is convex,
• f (x+1) = xf (x) for all x,
• f (1) =1. Then show that f (x) = [(x) for all x.

 

• State and prove Euler’s Theorem.

 

II  (a)  State and prove first version of Maximum Principle for Harmonic Functions.      (8)

OR

(b)  Let be a path from a to b and let be an analytic
continuation along There is a number Î>0 such that if  is any path
from a to b with  for all t, and if is any
contribution along  with [go]_a = [fo]_a : than prove that [g₁]_b = [f₁]_b .                   (8)

 

• (i) Define Poisson kernal and prove the four properties of Poisson kernal.

(ii)  Stating the required conditions, solve the Dirichlet’s problem for the unit disk.                                                                                                                     (6+11)

OR

(d)  State the prove Harnack’s inequality and hence prove Hernack’s theorem.           (5+12)

 

 

 

 

III  a.  State and prove Poisson – Jenson formula.                                                                 (8)

OR

1. State and prove Little Picard’s theorem.                                                                    (8)

1. Define order and genus of an entire function and prove that if f is an entire function

of finite genus then f is of finite order l £ +1.                                             (5+12)

 

OR

1. State and prove Bloch’s theorem.           (17)

 

IV  a. Prove that a discrete module consists of ether of Zero alone, of the integral multiples
nco of a single complex number ¹ o or  1 or of all  linear combinations  n₁ w₁ + h₂ w₂
with integral coefficient of two numbers w₁, w₂ with non real ration .                  (8)

OR

1. Show that the zeroes a₁, a₂ ….. a_n and poles b₁, b₂ ….b_n of an elliptic function satisfy (modm)

 

1. (i)  Define weierslvass p function.  Derive the differential equation satisfied by the

weierslvass p –function.

(ii)   Show that p (z +u) + p(z) + p (u)  =                          (2+8+7)

 

OR

1. Show that ℒ(z) is an odd function and prove that   ℒ(z) = -p(z), also derive the le gendre relation.

 

 

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