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(x) cos p. cd.e或I f(a sin prd.c 正确的做法是将被积函数取为f(z)ep2 如果函数f(2)e在上半平面内只有有限个奇
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 ({´òȼêf(z)eipz XJ¼êf(z)eipz3þ²¡SkkÛ : I §K C f(z)eipzdz = Z R −R f(x)eipxdx+ Z CR f(z)eipzdz = 2πi P þ²¡ res f(z)eipz 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(x) cos p. cd.e或I f(a sin prd.c 正确的做法是将被积函数取为f()ep2 如果函数f(2)e在上半平面内只有有限个奇 点,则 (a)eip=dz=/f(jeir-dz+/f()eip-adz 上半平面 cs{(9
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 ({´òȼêf(z)eipz XJ¼êf(z)eipz3þ²¡SkkÛ : I §K C f(z)eipzdz = Z R −R f(x)eipxdx+ Z CR f(z)eipzdz = 2πi P þ²¡ res f(z)eipz 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(x) cos p. cd.c或I f(a)sin p 如果f()满足 Jordan引理的要求,则有 f(z)e'prdz=2i 2 res f(a)eiaj 上半平面 分别比较实部和虚部,就可以求得 f(x)snpd(尖
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 XJf(z)÷vJordanÚn¦§Kk Z ∞ −∞ f(x)eipxdx = 2πi X þ²¡ res f(z)eipz ©O' Z ¢ÜÚJܧұ¦ ∞ −∞ f(x) cos pxdx Ú Z ∞ −∞ f(x) sin pxdx 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(x) cos p. cd.c或I f(a)sin p 如果f(z)满足 Jordan引理的要求,则有 f(r)elp-dz=2i > res (f(jeipa) 上半平面 分别比较实部和虚部,就可以求得 f(x) cos p. d.c和 f(x) ) sin pcd类
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 XJf(z)÷vJordanÚn¦§Kk Z ∞ −∞ f(x)eipxdx = 2πi X þ²¡ res f(z)eipz ©O' Z ¢ÜÚJܧұ¦ ∞ −∞ f(x) cos pxdx Ú Z ∞ −∞ f(x) sin pxdx C. S. Wu 1ù 3ê½n9ÙA^()
Evaluation of Definite Integrals(continued) sin c 例121计算积分 d r a>o 解】考虑复变积分 围道C如右图
Evaluation of Definite Integrals (continued) Integrals Involving Trigonometric Function ... Integrand with Singularity at Real Axis Integrals Involving Multivalued Functions ~12.1 OÈ© Z ∞ 0 x sin x x 2 + a 2 dx a > 0 =)> I ÄECÈ© C z z 2 + a 2 e iz dz CXmã I C z z 2+a 2 e iz dz = Z R −R x x 2+a 2 e ix dx + Z CR z z 2+a 2 e iz dz = 2πi res z z 2+a 2 e iz z=ia = 2πi · 1 2 e −a = πi e−a 4R → ∞ C. S. Wu 1ù 3ê½n9ÙA^()