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SOLVED MTH632 Assignment 1 Solution and Discussion

Fall 2019
MTH632: Complex Analysis and Differential Geometry
Assignment No. 1 (Lectures # 1 to 12) Total Marks: 10
Due Date: Tuesday, November 26, 2019Please read the following instructions before attempting the solution of this assignment:
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 No Assignment will be accepted once the Assignment solutions are uploaded.Question No. 1: Marks: 10
Suppose that f(z)=x^2y^22y+i(2x2xy), where z=x+iy. Use the expressions
x=(z+z ̅)/2 and y=(zz ̅)/2i
to write f(z) in terms of z, and simplify the result. 
Q. 1 Solution:
f(z)=x^2y^22y+i(2x2xy) f(z)=(z+z ̅ )^2/4+(zz ̅ )^2/4+i(zz ̅ )+i(z+z ̅ )(z+z ̅ )(zz ̅ )/2 f(z)=z^2/2+z ̅^2/2+2izz^2/2+z ̅^2/2 f(z)=z ̅^2+2iz  ANSWER

Q. 1 Solution:
f(z)=x^2y^22y+i(2x2xy) f(z)=(z+z ̅ )^2/4+(zz ̅ )^2/4+i(zz ̅ )+i(z+z ̅ )(z+z ̅ )(zz ̅ )/2 f(z)=z^2/2+z ̅^2/2+2izz^2/2+z ̅^2/2 f(z)=z ̅^2+2iz  ANSWER