Singular Point; Pole; 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.

Residue; Pole; Value of Integral; Counterclockwise

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.

Residue; Pole; Value of Integral; Counterclockwise

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'Hospital 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 factr that the value of this integral tends to zero as N tends to infinity, point out how it follows that: {see attachment} Please spe ...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.

Residues (Improper Integrals)

Use residues to evaluate the improper integrals (see attachment)

Residues; Cauchy Principal Values

Use residues to find the Cauchy principal values of the integrals {see attachment}

Residues (Improper Integrals)

Use residues to evaluate the improper integrals {see attachment}