Ch.6:Residue Thoory Ch.6:Residue Theory L6.3 Improper Integrals of Certain Fumctions Over() L6.3 Improper Integral of Certain Functions Over ( Improper Integrals of Certain Functions Over(-oo,oo) Improper Integrals of Certain Functions Over (-oo,oo) (Cont'd) Given any function f continuous on(-oo,oo).the limit P in f(a)ds Lemma If f(z)=P(z)/Q(z)is the quotient of two polynomials such that is called the Cauchy principal value of the integral of f over (-oo,oo),and we write degree Q≥2+degree P p.v.f(r)dz :lim then We shall now show how the theory of residue can be used to lim f(z)dz=0 compute p.v.integrals for certain functions of f See Example 1 on page 319 to learn the basic idea of the where C is the upper half-circle of radius p defined in Eq.(4)on algorithm page 320 as shown in Figure 6.4 Ch.6:Residue Thoory Ch.6:Residue Theory 6.3 Improper Integrals of Certain Functions Over ( 6.4 Improper Integras involving Trigonometric Functions Improper Integrals of Certain Functions Over(-oo,oo) Improper Integrals Involving Trigonometric Functions (Cont'd) The purpose of this section is to use residue theory to evaluate integrals of the general forms: P(r) r P(r) Then the improper integralf()dr can be computed as p.v. /-Q(x) cosmz dr,p.v. JoQ(r) sinmz dx follo If we obtain the value of the integral pv.felt=m2ai∑(inide） 00 P()eimz dr Jo Q(E) the above two integrals can be obtained by computing the real and imaginary parts +口·811定+1意1意00Ch.6: Residue Theory 6.3 Improper Integrals of Certain Functions Over (−∞, ∞) Improper Integrals of Certain Functions Over (−∞, ∞) Given any function f continuous on (−∞,∞), the limit lim ρ→∞ ρ−ρ f(x)dx is called the Cauchy principal value of the integral of f over (−∞,∞), and we write p.v. ∞−∞ f(x)dx := limρ→∞ ρ−ρ f(x)dx We shall now show how the theory of residue can be used to compute p.v. integrals for certain functions of f See Example 1 on page 319 to learn the basic idea of the algorithm Ch.6: Residue Theory 6.3 Improper Integrals of Certain Functions Over (−∞, ∞) Improper Integrals of Certain Functions Over (−∞, ∞) (Cont’d) Lemma If f(z) = P(z)/Q(z) is the quotient of two polynomials such that degree Q ≥ 2 + degree P then lim ρ→∞ C+ρ f(z)dz = 0 where C+ρ is the upper half-circle of radius ρ defined in Eq. (4) on page 320 as shown in Figure 6.4 Ch.6: Residue Theory 6.3 Improper Integrals of Certain Functions Over (−∞, ∞) Improper Integrals of Certain Functions Over (−∞, ∞) (Cont’d) Then the improper integral  ∞−∞ f(x)dx can be computed as follows p.v. ∞−∞ f(x)dx = limρ→∞ 2πi(residues inside Γρ) Ch.6: Residue Theory 6.4 Improper Integrals Involving Trigonometric Functions Improper Integrals Involving Trigonometric Functions The purpose of this section is to use residue theory to evaluate integrals of the general forms: p.v. ∞−∞ P(x) Q(x) cos mx dx, p.v. ∞−∞ P(x) Q(x) sin mx dx If we obtain the value of the integral ∞−∞ P(x) Q(x)eimx dx the above two integrals can be obtained by computing the real and imaginary parts