1. 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.
2. 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.
Summation (going from 0 to infinity) 1/n^2 +a^2
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.
5. Evaluate the following principal value integral using an appropriate contour.
Integration of (integral goes from 0 to infinity) : (x)^a-1/1-x^2,
This solution is comprised of a detailed explanation to 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.