Chapter 7 Applications of Residues - evaluation of definite and improper integrals occurring in real analysis and applied math - finding inverse Laplace transform by the methods of summing residues. 60. Evaluation of Improper Integrals In calculus when the limit on the right exists, the improper integral is said to converge to that limit.

If f(x) is continuous for all x, its improper integral over the is defined by When both of the limits here exist, integral (2) converges to their sum. There is another value that is assigned to integral (2). i.e., The Cauchy principal value (P.V.) of integral (2). provided this single limit exists

If integral (2) converges its Cauchy principal value (3) exists. If is not, however, always true that integral (2) converges when its Cauchy P.V. exists. Example. (ex8, sec. 60)

(1) (3) if exist

To evaluate improper integral of p, q are polynomials with no factors in common. and q(x) has no real zeros. See example

Example has isolated singularities at 6th roots of –1. and is analytic everywhere else. Those roots are

61. Improper Integrals Involving sines and cosines To evaluate Previous method does not apply since sinhay (See p.70) However, we note that

Ex1. An even function And note that is analytic everywhere on and above the real axis except at

Take real part

It is sometimes necessary to use a result based on Jordan’s inequality to evaluate

Suppose f is analytic at all points

Example 2. Sol:

But from Jordan’s Lemma

62. Definite Integrals Involving Sines and Cosines

63. Indented Paths

Ex1. Consider a simple closed contour

Jordan’s Lemma

64. Integrating Along a Branch Cut (P.81, complex exponent)

Then

65. Argument Principle and Rouche’s Theorem A function f is said to be meromorphic in a domain D if it is analytic throughout D - except possibly for poles. Suppose f is meromorphic inside a positively oriented simple close contour C, and analytic and nonzero on C. The image of C under the transformation w = f(z), is a closed contour, not necessarily simple, in the w plane.

Positive: Negative:

The winding number can be determined from the number of zeros and poles of f interior to C. Number of poles zeros are finite (Ex 15, sec. 57) (Ex 4) Argument principle

Pf.

Rouche’s theorem Thm 2. Pf.

66. Inverse Laplace Transforms Suppose that a function F of complex variable s is analytic throughout the finite s plane except for a finite number of isolated singularities. Bromwich integral

Jordan’s inequality

67. Example Exercise 12 When t is real

Ex1.