Chapter 7 Applications of Residues evaluation of definite

• If f(x) is continuous for all x, its improper integral over the

If integral (2) converges its Cauchy principal value (3) exists. If is not, however,

• To evaluate improper integral of p, q are polynomials with no factors

Example has isolated singularities at 6 th roots of – 1. and is analytic

61. Improper Integrals Involving sines and cosines To evaluate Previous method does not apply

Ex 1. An even function And note that is analytic everywhere on and above

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

62. Definite Integrals Involving Sines and Cosines 17

Ex 1. Consider a simple closed contour 22

64. Integrating Along a Branch Cut (P. 81, complex exponent) 24

65. Argument Principle and Rouche’s Theorem A function f is said to be meromorphic

The winding number can be determined from the number of zeros and poles of

66. Inverse Laplace Transforms Suppose that a function F of complex variable s is

67. Example Exercise 12 When t is real 43

Slides: 46

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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. 1

• 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 2

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. (ex 8, sec. 60) 3

(1) (3) if exist 4

• To evaluate improper integral of p, q are polynomials with no factors in common. and q(x) has no real zeros. See example 5

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

61. Improper Integrals Involving sines and cosines To evaluate Previous method does not apply since sinhay (See p. 70) However, we note that 9

Ex 1. An even function And note that is analytic everywhere on and above the real axis except at 10

Take real part 11

It is sometimes necessary to use a result based on Jordan’s inequality to evaluate 12

Suppose f is analytic at all points 13

Example 2. Sol: 14

But from Jordan’s Lemma 16

62. Definite Integrals Involving Sines and Cosines 17

63. Indented Paths 20

Ex 1. Consider a simple closed contour 22

Jordan’s Lemma 23

64. Integrating Along a Branch Cut (P. 81, complex exponent) 24

Then 26

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. 28

Positive: Negative: 30

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 31

Pf. 32

Rouche’s theorem Thm 2. Pf. 36

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 39

Jordan’s inequality 41

67. Example Exercise 12 When t is real 43

Ex 1. 44