Chapter 7. Applications of Residues Weiqi Luo ( 骆伟祺 ) School of Software Sun Yat-Sen University ： Office ： # A313

School of Software  Evaluation of Improper Integrals  Improper Integrals From Fourier Analysis  Jordan’s Lemma  Definite Integrals Involving Sines and Cosines 2 Chapter 7: Applications of Residues

School of Software  Improper Integral 78. Evaluation of Improper Integrals 3 If f is continuous for the semi-infinite interval 0≤x<∞ or all x, its improper integrals are defined as when the limit/limits on the right exists, the improper integral is said to converge to that limit/their sum.

School of Software  Cauchy Principal Value (P.V.) 78. Evaluation of Improper Integrals 4

School of Software  Example Observe that However, 78. Evaluation of Improper Integrals 5 An Odd Function No limits

School of Software  Suppose f(x) is an even function and assume that the Cauchy principal value exists, then 78. Evaluation of Improper Integrals 6

School of Software  Evaluation Improper Integrals of Ration Functions where p(x) and q(x) are polynomials with real coefficients and no factors in common. assume that q(z) has no real zeros but at least one zero above the real axis, labeled z 1, z 2, …, z n, where n is less than or equal to the degree of q(z) 78. Evaluation of Improper Integrals 7

School of Software 78. Evaluation of Improper Integrals 8 When When f(x) is even

School of Software  Properties 79. Example 9 Let where m≥n+2, a n ≠0, b m ≠0, then we get ∞ 0

School of Software  Example 79. Example 10 Firstly, find the roots of the function None of them lies on the real axis, and the first three roots lie in the upper half plane And 6-2=4≥2

School of Software  Example(Cont’) 79. Example 11 Here the points c k are simple poles of f, according to the Theorem 2 in pp. 253, we get that Even Function

School of Software pp. 267 Ex. 3, Ex. 4, Ex. 7, Ex Homework 12

School of Software  Improper Integrals of the Following Forms 80. Improper Integrals From Fourier Analysis 13 OR where a denotes a positive constant where p(x) and q(x) are polynomials with real coefficients and no factors in common. Also, q(x) has no zeros on the real axis and at least one zero above it.

School of Software  Improper Integrals In Sec. 78 & 79, 80. Improper Integrals From Fourier Analysis 14 The moduli increase as y tends to infinity This moduli is bounded in the upper plane y>0 (a>0), and is larger than 0.

School of Software  Example Let us show that Because the integrand is even, it is sufficient to show that the Cauchy principal value of the integral exists and to find that value. We introduce the function The product f(z)e i3z is analytic everywhere on and above the real axis except at the point z=i. 80. Improper Integrals From Fourier Analysis 15

School of Software  Example (Cont’) 80. Improper Integrals From Fourier Analysis 16 the point z = i is evidently a pole of order m = 2 of f (z)e i3z, and

School of Software  Example (Cont’) 80. Improper Integrals From Fourier Analysis 17  0, when R  ∞

School of Software  Theorem Suppose that a)a function f (z) is analytic at all points in the upper half plane y ≥ 0 that are exterior to a circle |z| = R 0 ; b)C R denotes a semicircle z = Re iθ (0 ≤ θ ≤ π), where R > R 0 ; c)for all points z on C R, there is a positive constant M R such that 81. Jordan’s Lemma 18 Then, for every positive constant a,

School of Software 81. Jordan’s Lemma 19 The Jordan’s Inequality Consider the following two functions sinΘ is symmetric with Θ=π/2

School of Software 81. Jordan’s Lemma 20 According to the Jordan’s Inequation, it follows that The final limit in the theorem is now evident since M R  0 as R  ∞

School of Software  Example Let us find the Cauchy principal value of the integral we write where z 1 =-1+i. The point z 1, which lies above the x axis, is a simple pole of the function f(z)e iz, with residue 81. Jordan’s Lemma 21

School of Software  Example (Cont’) which means that 81. Jordan’s Lemma 22

School of Software  Example (Cont’) 81. Jordan’s Lemma 23 0 However, based on the Theorem, we obtain that Since

School of Software pp Ex. 2, Ex. 4, Ex. 9, Ex Jordan’s Lemma 24

School of Software  Evaluation of the Integrals 85. Definite Integrals Involving Sines and Cosines 25 The fact that θ varies from 0 to 2π leads us to consider θ as an argument of a point z on a positively oriented circle C centered at the origin. Taking the radius to be unity C, we use the parametric representation

School of Software  Example Let us show that 85. Definite Integrals Involving Sines and Cosines 26 where C is the positively oriented circle |z|=1.

School of Software  Example (Cont’) 85. Definite Integrals Involving Sines and Cosines 27 Hence there are no singular points on C, and the only one interior to it is the point z 1. The corresponding residue B 1 is found by writing This shows that z 1 is a simple pole and that

School of Software  Example (Cont’) 85. Definite Integrals Involving Sines and Cosines 28

School of Software pp Ex. 1, Ex. 3, Ex Definite Integrals Involving Sines and Cosines 29