Evaluation of Definite Integrals(continued) Jordan引理 (要点) 设在0≤ag之≤π范围内,当 12→∞时Q()→0,则 Q(=)e"dz=0 R→∞JCR 【证】当z在CR上时,z=Re Q()e"pd l Q(Re)e-pR sin Rde ≤ER/er2md=2R e"pRsin g 证明的关键在于精确估计sinθ值
Evaluation of Definite Integrals (continued) Integrals Involving Trigonometric Function ... Integrand with Singularity at Real Axis Integrals Involving Multivalued Functions JordanÚn (:) 30≤arg z≤πS§ |z|→∞Q(z)⇒0§K lim R→∞ Z CR Q(z)eipzdz = 0 =y>z3CRþ§z = Re iθ Z CR Q(z)eipzdz ≤ Z π 0 Q(Re iθ ) e −pR sin θRdθ ≤εR Z π 0 e −pR sin θ dθ =2εR Z π/2 0 e −pR sin θ dθ y²' 3u°(Osin θ C. S. Wu 1ù 3ê½n9ÙA^()
Evaluation of Definite Integrals(continued) (=)ed sin e <2R Rsin e 2=R 当0≤6≤丌/2时 sine≥20/π
Evaluation of Definite Integrals (continued) Integrals Involving Trigonometric Function ... Integrand with Singularity at Real Axis Integrals Involving Multivalued Functions 0 ≤ θ ≤ π/2 sin θ ≥ 2θ/π Z CR Q(z)eipzdz ≤ 2εR Z π/2 0 e −pR sin θ dθ ≤ 2εR Z π/2 0 e −pR·2θ/π dθ = 2εR · π 2pR 1 − e −pR = επ p 1 − e −pR → 0 ∴ lim R→∞ Z CR Q(z)eipzdz = 0 C. S. Wu 1ù 3ê½n9ÙA^()
Evaluation of Definite Integrals(continued) Q()e'p2dzl sin e <2R Rsin e 2 <2R 当0≤6≤丌/2时 sine≥20/π
Evaluation of Definite Integrals (continued) Integrals Involving Trigonometric Function ... Integrand with Singularity at Real Axis Integrals Involving Multivalued Functions 0 ≤ θ ≤ π/2 sin θ ≥ 2θ/π Z CR Q(z)eipzdz ≤ 2εR Z π/2 0 e −pR sin θ dθ ≤ 2εR Z π/2 0 e −pR·2θ/π dθ = 2εR · π 2pR 1 − e −pR = επ p 1 − e −pR → 0 ∴ lim R→∞ Z CR Q(z)eipzdz = 0 C. S. Wu 1ù 3ê½n9ÙA^()
Evaluation of Definite Integrals(continued) (a)epad <2R Rsin e sin e 2 <2:R 当0≤6≤丌/2时 sine≥20/π
Evaluation of Definite Integrals (continued) Integrals Involving Trigonometric Function ... Integrand with Singularity at Real Axis Integrals Involving Multivalued Functions 0 ≤ θ ≤ π/2 sin θ ≥ 2θ/π Z CR Q(z)eipzdz ≤ 2εR Z π/2 0 e −pR sin θ dθ ≤ 2εR Z π/2 0 e −pR·2θ/π dθ = 2εR · π 2pR 1 − e −pR = επ p 1 − e −pR → 0 ∴ lim R→∞ Z CR Q(z)eipzdz = 0 C. S. Wu 1ù 3ê½n9ÙA^()
Evaluation of Definite Integrals(continued) (=)ed sin e <2R Rsin e 2 <2:R 当0≤6≤丌/2时 1-e-pn)0 sine≥20/π
Evaluation of Definite Integrals (continued) Integrals Involving Trigonometric Function ... Integrand with Singularity at Real Axis Integrals Involving Multivalued Functions 0 ≤ θ ≤ π/2 sin θ ≥ 2θ/π Z CR Q(z)eipzdz ≤ 2εR Z π/2 0 e −pR sin θ dθ ≤ 2εR Z π/2 0 e −pR·2θ/π dθ = 2εR · π 2pR 1 − e −pR = επ p 1 − e −pR → 0 ∴ lim R→∞ Z CR Q(z)eipzdz = 0 C. S. Wu 1ù 3ê½n9ÙA^()