Complex Variables : Analytic Functions and Limits

Let f(z) be analytic in a region G and set φ(z,w) = (f(w)-f(z))/(w-z) for w,z Є G w ≠ z. Let z0 Є G. Show that lim (z,w)-->(z0,z0) φ(z,w) =f'(z0). Complex Variables. See attached file for full problem description

Complex Variables : Rectafiable Curve

See attached file for full problem description.

Complex Variables : Rectafiable Curves

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).

Compact Sets and Compact Exhaustions

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

Singularities and Poles

The function f(z) = zsin(pi/z)/[(z-1)(z-2)^2] has isolated singularities only. Determine the singularities of f(z) and classify each of them as removable, a pole, or an essential singularity. If z0 is a removable singularity, find the value f(zo) that makes f(z) analytic at z0. If z0 is a pole. find the singular part of f(z) at ...continues

Complex Variables : Taylor Polynomial Proof

Please see the attached file for the fully formatted problems.

Meromorphic Functions

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

Mobius Transformations

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 Transformations

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.