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 improper integrals {see attachmnet}

Derive Integration Formula; Trigonometric Identity

Derive the attached integration formula. Then, with the aid of the trigonometric identity {see attachment} point out how it follows that ... {see attachment} **See attachment for complete question including diagram.

Pair of Integration Functions

Use the attached function to derive the pair of integration functions {see attached for functions and diagram}

Complex Variable Class - Undergraduate 500 Level

Show that... by integrating an appropriate branch of the multiple-valued function... over (a) the indented path in Fig. 97, Sec 75; (b) the closed contour in Fig. 99, Sec. 77. See attachment for equations and diagrams.

Complex Variable Class - Undergraduate 500 Level

The beta function is this function of two real variables... Make the substitution t=1/(x+1) and use the result obtained in the example in Sec. 77 to show that... Please see attachment for equations.

Complex Variable Class - Undergraduate 500 Level

Please try to use the example method. 1. F(s) = (2s^3)/((s^4) - 4) Please see attachment for proper format.

Complex Variable Class - Undergraduate 500 Level

Use the formal method, involving an infinite series of residues and illustrated in the examples. 7. F(s) = 1/(s cosh (s^½)) Please see attachment for full question.

Complex Variable Class - Undergraduate 500 Level

Suppose that a function F(s) has a pole of order m at s=s0, with a LAurent series expansion ... in the punctured disk 0<|s - s0| ...continues

It is an explanation of the Cauchy-Hadamard theorem on power series. Cauchy- Hadamard Theorem :- ∞ For every power series ∑ anzn there exist a number R, 0 ≤ R < ∞ n=0 called the radius of convergence with the following properties: (i) The series converges absolutely for every |z| < R (ii) If 0 ≤ ρ < R, the series converges uniformly for |z| ≤ ρ (iii) If |z| > R, the terms of the series are unbounded and the series is consequently divergent.

Complex Variable Cauchy- Hadamard Theorem :- ...continues