Evaluation of Definite Integrals(continued) (=)ed <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 sine≥20/π im/?(=)edz=0口 →o
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^()
含三角函数的无穷积分 这类积分的标准形式 (不妨设p>0) f(a)cos p.dr x I=/ f(a)sin prdr 处理这种类型的积分,仍可以采用半圆形的围道 被积函数不能简单地取为f(2)c0sp或f(=)snp2
Evaluation of Definite Integrals (continued) Integrals Involving Trigonometric Function ... Integrand with Singularity at Real Axis Integrals Involving Multivalued Functions ¹n¼êáȩ ùaÈ©IO/ª (Øp > 0) I = Z ∞ −∞ f(x) cos pxdx ½ I = Z ∞ −∞ f(x) sin pxdx ?nù«a.È©§E±æ^ / ȼêØU{ü/f(z) cos pz½f(z) sin pz ϵØBuO lim R→∞ Z CR f(z) cos pzdz ½ lim R→∞ Z CR f(z) sin pzdz C. S. Wu 1ù 3ê½n9ÙA^()
含三角函数的无穷积分 这类积分的标准形式 (不妨设p>0) f(a)cos p.dr x I=/ f(a)sin prdr 处理这种类型的积分,仍可以采用半圆形的围道 被积函数不能简单地取为f()cosp或f(z) sin pz 原因:不便于直接计算 () cos pzd2或mf(=) 尜
Evaluation of Definite Integrals (continued) Integrals Involving Trigonometric Function ... Integrand with Singularity at Real Axis Integrals Involving Multivalued Functions ¹n¼êáȩ ùaÈ©IO/ª (Øp > 0) I = Z ∞ −∞ f(x) cos pxdx ½ I = Z ∞ −∞ f(x) sin pxdx ?nù«a.È©§E±æ^ / ȼêØU{ü/f(z) cos pz½f(z) sin pz ϵØBuO lim R→∞ Z CR f(z) cos pzdz ½ lim R→∞ Z CR f(z) sin pzdz C. S. Wu 1ù 3ê½n9ÙA^()
含三角函数的无穷积分 这类积分的标准形式 (不妨设p>0) f(a)cos p.dr x I=/ f(a)sin prdr 处理这种类型的积分,仍可以采用半圆形的围道 被积函数不能简单地取为f()cosp或f(z) sin pz 原因:不便于直接计算 im/f() cos pad或Ii f(a)sin pzd R→∞o
Evaluation of Definite Integrals (continued) Integrals Involving Trigonometric Function ... Integrand with Singularity at Real Axis Integrals Involving Multivalued Functions ¹n¼êáȩ ùaÈ©IO/ª (Øp > 0) I = Z ∞ −∞ f(x) cos pxdx ½ I = Z ∞ −∞ f(x) sin pxdx ?nù«a.È©§E±æ^ / ȼêØU{ü/f(z) cos pz½f(z) sin pz ϵØBuO lim R→∞ Z CR f(z) cos pzdz ½ lim R→∞ Z CR f(z) sin pzdz C. S. Wu 1ù 3ê½n9ÙA^()