Ch.6:Residue Thoory Ch.6:Residue Theory 6.4Imprope Integrals Involving Trigonone Functions Improperntegra Involving Trigonometric Functions Improper Integrals Involving Trigonometric Functions Improper Integrals Involving Trigonometric Functions (Cont'd) (Cont'd) Lemma Then the improper integralf()dr can be computed as If m>0 and P/Q is the quotient of two polynomials such that follows degree Q≥1+degree P 广em号女=▣2a∑(e Q(z) then Thus mf eims P(d cos m Q( Q() P-c+Q(r) where C is the upper half-circle of radius p = sin m Q() 白·0+之。·急，是20CCh.6: Residue Theory 6.4 Improper Integrals Involving Trigonometric Functions Improper Integrals Involving Trigonometric Functions (Cont’d) Lemma If m > 0 and P/Q is the quotient of two polynomials such that degree Q ≥ 1 + degree P then lim ρ→∞ C+ρ eimxP(x) Q(x)dz = 0 where C+ρ is the upper half-circle of radius ρ Ch.6: Residue Theory 6.4 Improper Integrals Involving Trigonometric Functions Improper Integrals Involving Trigonometric Functions (Cont’d) Then the improper integral  ∞−∞ f(x)dx can be computed as follows p.v. ∞−∞ eimxP(x) Q(x)dx = limρ→∞ 2πi(residues inside Γρ) Thus p.v. ∞−∞ cos mxP(x) Q(x)dx =  p.v. ∞−∞ eimxP(x) Q(x)dx p.v. ∞−∞ sin mxP(x) Q(x)dx =  p.v. ∞−∞ eimxP(x) Q(x)dx