Maclaurin Series : Circles

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 ...continues

Isolated Singluar Point: Pole, Removable Single Point and 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.

Singular Point : Pole and Residue

2. Show that the singular point of each of the following functions is a pole. Determine the order m of that pole and the corresponding residue B. {please see attachment for functions} Please specify the terms that you use if necessary and clearly explain each step of your solution.

Residues and Poles; Polar Numbers; Demoivre's Theorem

Please see the attachment for questions relating to residues and poles (polar numbers and Demoivre's Theorem). These problems are from complex variable class. Please specify the terms that you use if necessary and clearly explain each step of your solution.

Residues and Poles : Cauchy Integral Formula (Counterclockwise around a Circle)

Find the value of the integral: {see attachment} taken counterclockwise around the circle (a) |z - 2| = 2 (b) |z| = 4 Please specify the terms that you use if necessary and clearly explain each step of your solution.

Residues and Poles : Value of Integral (Counterclockwise around a Circle)

Fine the value of the integral {see attachment} taken counterclockwise around the circle: (a) |z| = 2 (b) |z + 2| = 3 Please specify the terms that you use if necessary and clearly explain each step of your solution.

Residues and Poles : L'Hopital's Rule

See attachment for questions relating to residues and poles. Please specify the terms that you use if necessary and clearly explain each step of your solution.

Residues and Poles : Positively Oriented Boundary

5. Let Cn denote the positively oriented boundary of the square whose edges lie along the lines: {see attachment}, where N is a positive integer. Show that {see attachment}. then, using the factor that the value of this integral tends to zero as N tends to infinity, point out how it follows that: {see attachment} Please sp ...continues

Residues and Poles : Positively Oriented Boundary

See attachment for question relating to residues and poles. Please specify the terms that you use if necessary and clearly explain each step of your solution.