Complex Variables - Taylor Series

This problem is from complex variable class. Please specify the terms that you use if necessary and clearly explain each step of your solution. Problem: Show that when ... (see attachment!)

Find Residue; Laurent Series

1. Find the residue at z = 0 of the function: {see attachment} Please specify the terms that you use if necessary and clearly explain each step of your solution.

Maclaurin Series

4. Let C denote the circle |z|=1, taken counterclockwise, and following the steps below to show that: {see attachment for steps and equation} Please specify the terms that you use if necessary and clearly explain each step of your solution.

Single Residue; Interior to Closed Contour

5. Let the degress of the polynomials {see attachment} be such that m [less than or equal to] n+2. Use the theorem in Sec. 64 {see attachment} to show that if all of the zeros of Q(z) are interior to a simple closed contour C, then {see attachment} Please specify the terms that you use if necessary and clearly explain each step of your solution.

Isolated Singluar Point: Pole, Removable Single Point, Essential Single Point

1. In each case, write the principal part of the function at its isolated singular point and determine whether that point it a pole, a removable single point, or an essential singular point {see attachment for expressions} Please specify the terms that you use if necessary and clearly explain each step of your solution.