Assignment:
Q1. A function f(z) is said to be periodic with a period a, a is not equal to zero.
if f(z+ma) =f(z),
where m is an integer different from zero. prove that a function, which has two distinct periods say, a and b which are not integer multiples of the other- can not be regular in the entire complex plane.
Note: Doubly periodic functions, called elliptic functions have been constructed.
Q2. Let f(z) be an analytic function of the complex variable z on a domain D. Let C be a smooth closed curve inside D and suppose that C and its interior E are mapped on to the unit disc /w/<=1. Prove that the points on the boundary /w/=1 can not be the image of an interior point of E.
Q3. Using contour integration and calculus of residues, find the sum
Summation (going from 0 to infinity) 1/n^2 +a^2
Q4. Let
Sinz = z-z^3/3! +z^5/5!-----------+ (-1)^2n+1(z)^2n+1/(2n+1)!
Prove that the function sinz/z has an infinite number of zeros.
Q5. Evaluate the following principal value integral using an appropriate contour.
Integration of (integral goes from 0 to infinity) : (x)^a-1/1-x^2,
0
Provide complete and step by step solution for the question and show calculations and use formulas.