Fall 2019
MTH632: Complex Analysis and Differential Geometry
Assignment No. 1 (Lectures # 1 to 12) Total Marks: 10
Due Date: Tuesday, November 26, 2019
Please read the following instructions before attempting the solution of this assignment:
To solve this assignment, you should have good command over 1 to 12 lectures.
Try to consolidate your concepts that you learn in the lectures with these questions.
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Question No. 1: Marks: 10
Suppose that f(z)=x^2-y^2-2y+i(2x-2xy), where z=x+iy. Use the expressions
x=(z+z ̅)/2 and y=(z-z ̅)/2i
to write f(z) in terms of z, and simplify the result.
Q. 1 Solution:
f(z)=x^2-y^2-2y+i(2x-2xy)
f(z)=(z+z ̅ )^2/4+(z-z ̅ )^2/4+i(z-z ̅ )+i(z+z ̅ )-(z+z ̅ )(z-z ̅ )/2
f(z)=z^2/2+z ̅^2/2+2iz-z^2/2+z ̅^2/2
f(z)=z ̅^2+2iz - ANSWER
Q. 1 Solution:
f(z)=x^2-y^2-2y+i(2x-2xy)
f(z)=(z+z ̅ )^2/4+(z-z ̅ )^2/4+i(z-z ̅ )+i(z+z ̅ )-(z+z ̅ )(z-z ̅ )/2
f(z)=z^2/2+z ̅^2/2+2iz-z^2/2+z ̅^2/2
f(z)=z ̅^2+2iz - ANSWER
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