Complex Variables

Hint: show that σ1 ~ σ2 and then use the transitivity of ~, where ~ means homotopic to i.e. that If γ1 ~ γ2 then there exists a map Г:[0,1]x[0,1]  G, where G is a region in C, such that Г(t,0) = γ1 Г(t,1) = γ2 And Г(0,s) = Г(1,s) for all s in [0,1] For ...continues

Complex Variables

There are two problems in the file - this post is for the second problem, on the second page, where you are asked to evaluate the integral for all positive n. See attached file for full problem description.

Radius of Convergence of Complex Power Series

I have to find the radius of convergence for the following series: Sum from j = 0 to infinity of z^(3j)/2^j, and the answer according to my book is R = 2^(1/3). Can someone please provide details on how this solution comes about.

Topology on complex plane

Definition: Let omega be a domain in C. Then e compact exhaustion {Ek} of omega is 1. Ek are all compact, Ek is contained in Ek+1 for all k 2. Union of Ek=omega 3. Any compact set K contained in omega is contained in some Ek Problem. Find an example of Ek's satisfying 1 and 2 but not 3 for omega=unit disk

Determine the singularities

See attached file for full problem description.

Complex Variables

Complex Variables. I think you use Rouche's thm. with e^z and e^z - Pn(z) but I'm not sure how. See attached file for full problem description.

Meromorphic Function

Show that any function which meromorphic in the extended plane is rational.

Mobius Transformation proof

Prove that if T is a Mobius transformation such that T(0) = 0, then T may be written as T(z) = z/(cz+d) for some choices of c and d.

Mobius transformation proof

Suppose T is a Mobius transformation such that the image of the real axis under T is the real axis. Prove that T may be written in the form T(z) = (az+b)/(cz+d) with a, b, c, and d real.

Principal branch complex powers

How do you find (and what is it) the principal branch of the following complex powers: (e/2)(1-i(sqrt(3)))^(3pi*i) and (1-i)^(4i)