Evaluation of Definite Integrals(continued) sin c 例121计算积分 d r a>o 【解】考虑复变积分 2⊥n2edz 围道C如右图 ruIe
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^()
Evaluation of Definite Integrals(continued) sin c 例121计算积分 d r a>o 【解】考虑复变积分 2⊥n2edz 围道C如右图 +a2e de R T ed x z o dz r+ +a =2i res 22+a2 27t.-
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^()
Evaluation of Definite Integrals(continued) sin c 例121计算积分 d r a>o 【解】考虑复变积分 2⊥n2edz 围道C如右图 +a2e de R T ed x z o dz r+ Cp 2+as =2i res 22+a2 27
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^()
Evaluation of Definite Integrals(continued) sin c 例121计算积分 d r a>o 【解】考虑复变积分 2⊥n2edz 围道C如右图 +a2e de R T ed x z o dz r+ +a =2i res 22+a2 27 27ie-a 取极限R
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^()
Evaluation of Definite Integrals(continued) sin c 例121计算积分 d r a>o 【解】考虑复变积分 2⊥n2edz 围道C如右图 +a2e de R T ed x z o dz r+ +a 2TTi res 27 27ie-a 取极限R→0
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^()