Outline1ù3ê½n9ÙA^()ÔnÆêÆÔn{§|2007cSC.S.Wu1ù3ê½n9ÙA^()OutlineùÇ:13ê½n3ê½n3ê½nÐÚA^á:?3ê23ê½nO½È©knn¼êȩáȩC.S.Wu1ù3ê½n9ÙA^()OutlineùÇ:13ê½n3ê½n3ê½nÐÚA^á:?3ê23ê½nO½È©knn¼êȩáȩC.S.Wu1ù3ê½n9ÙA^()ResiduetheoremEvaluationofDefiniteIntegralsReferencesÇÂÁ§5êÆÔn{6§§7.1—7.3ù&§5êÆÔn{6§§4.1,4.2nÎ!X1Á§5êÆÔn{6§§5.1,5.2,5.3C.S.Wu1ù3ê½n9ÙA^()ResiduetheoremEvaluationofDefiniteIntegralsResiduetheoremSomeApplicationsofResidueTheoremResidueatInfinityùÇ:13ê½n3ê½n3ê½nÐÚA^á:?3ê23ê½nO½È©knn¼êȩáȩC.S.Wu1ù3ê½n9ÙA^()ResiduetheoremEvaluationofDefiniteIntegralsResiduetheoremSomeApplicationsofResidueTheoremResidueatInfinity3êÚ\3/«R1≤|z−b|≤R2Sü)Û¼êf(z)§±3T«SÐm¤Laurent?êf(z)=∞Pn=−∞an(z−b)nR1|z−b|R2an=12πiICf(ζ)(ζ−b)n+1dζù´Laurent?ê½Â©¢^þé|^ù½ÂOÐmXê§õ¹elOå»N´¦ÑXêC.S.Wu1ù3ê½n9ÙA^()ResiduetheoremEvaluationofDefiniteIntegralsResiduetheoremSomeApplicationsofResidueTheoremResidueatInfinity3êÚ\3/«R1≤|z−b|≤R2Sü)Û¼êf(z)§±3T«SÐm¤Laurent?êf(z)=∞Pn=−∞an(z−b)nR1|z−b|R2an=12πiICf(ζ)(ζ−b)n+1dζù´Laurent?ê½Â©¢^þé|^ù½ÂOÐmXê§õ¹elOå»N´¦ÑXêC.S.Wu1ù3ê½n9ÙA^()ResiduetheoremEvaluationofDefiniteIntegralsResiduetheoremSomeApplicationsofResidueTheoremResidueatInfinity3êÚ\3/«R1≤|z−b|≤R2Sü)Û¼êf(z)§±3T«SÐm¤Laurent?êf(z)=∞Pn=−∞an(z−b)nR1|z−b|R2an=12πiICf(ζ)(ζ−b)n+1dζù´Laurent?ê½Â©¢^þé|^ù½ÂOÐmXê§õ¹elOå»N´¦ÑXêC.S.Wu1ù3ê½n9ÙA^()ResiduetheoremEvaluationofDefiniteIntegralsResiduetheoremSomeApplicationsofResidueTheoremResidueatInfinity3êÚ\L5A^þª§ÒU^5OÈ©ICf(ζ)(ζ−b)n+1dζ=2πianAO´§n=−1ICf(ζ)dζ=2πia−13,¹e§kUN´¦a−1ùÒ´3ê½nÄgC.S.Wu1ù3ê½n9ÙA^()ResiduetheoremEvaluationofDefiniteIntegralsResiduetheoremSomeApplicationsofResidueTheoremResidueatInfinity3êÚ\L5A^þª§ÒU^5OÈ©ICf(ζ)(ζ−b)n+1dζ=2πianAO´§n=−1ICf(ζ)dζ=2πia−13,¹e§kUN´¦a−1ùÒ´3ê½nÄgC.S.Wu1ù3ê½n9ÙA^()ResiduetheoremEvaluationofDefiniteIntegralsResiduetheoremSomeApplicationsofResidueTheoremResidueatInfinity3êÚ\L5A^þª§ÒU^5OÈ©ICf(ζ)(ζ−b)n+1dζ=2πianAO´§n=−1ICf(ζ)dζ=2πia−13,¹e§kUN´¦a−1ùÒ´3ê½nÄgC.S.Wu1ù3ê½n9ÙA^()ResiduetheoremEvaluationofDefiniteIntegralsResiduetheoremSomeApplicationsofResidueTheoremResidueatInfinity3ê½n«G.C©ã1w{ü4Ü©eØkáÛ:bk,k=1,2,3,···,n §¼êf(z)3GSü)Û§3G¥ëY§ 3Cþvkf(z)Û:§KICf(z)dz=2πinXk=1resf(bk)resf(bk)¡f(z)3bk?3ꧧuf(z)3bkSLaurentÐm¥(z−bk)−1Xêa(k)−1C.S.Wu1ù3ê½n9ÙA^()ResiduetheoremEvaluationofDefiniteIntegralsResiduetheoremSomeApplicationsofResidueTheoremResidueatInfinity3ê½n(:)ICf(z)dz=2πinXk=1resf(bk)=y7zÛ:bk4Üγk§¦γkþ3GS§ pØUKâEëÏ«Cauchy½n9LaurentÐmXêúª§=ICf(z)dz=nPk=1Iγkf(z)dz=2πinPk=1a(k)−1=2πinPk=1resf(bk)C.S.Wu1ù3ê½n9ÙA^()ResiduetheoremEvaluationofDefiniteIntegralsResiduetheoremSomeApplicationsofResidueTheoremResidueatInfinity3ê½n(:)ICf(z)dz=2πinXk=1resf(bk)=y7zÛ:bk4Üγk§¦γkþ3GS§ pØUKâEëÏ«Cauchy½n9LaurentÐmXêúª§=ICf(z)dz=nPk=1Iγkf(z)dz=2πinPk=1a(k)−1=2πinPk=1resf(bk)C.S.Wu1ù3ê½n9ÙA^()ResiduetheoremEvaluationofDefiniteIntegralsResiduetheoremSomeApplicationsofResidueTheoremResidueatInfinity3ê½n(:)ICf(z)dz=2πinXk=1resf(bk)=y7zÛ:bk4Üγk§¦γkþ3GS§ pØUKâEëÏ«Cauchy½n9LaurentÐmXêúª§=ICf(z)dz=nPk=1Iγkf(z)dz=2πinPk=1a(k)−1=2πinPk=1resf(bk)C.S.Wu1ù3ê½n9ÙA^()ResiduetheoremEvaluationofDefiniteIntegralsResiduetheoremSomeApplicationsofResidueTheoremResidueatInfinity3ê½n(:)ICf(z)dz=2πinXk=1resf(bk)=y7zÛ:bk4Üγk§¦γkþ3GS§ pØUKâEëÏ«Cauchy½n9LaurentÐmXêúª§=ICf(z)dz=nPk=1Iγkf(z)dz=2πinPk=1a(k)−1=2πinPk=1resf(bk)C.S.Wu1ù3ê½n9ÙA^()ResiduetheoremEvaluationofDefiniteIntegralsResiduetheoremSomeApplicationsofResidueTheoremResidueatInfinityµã3ê½n¢3ê½n=áÛ:Vg+CauchyÈ©úª+LaurentÐmXêúªC.S.Wu1ù3ê½n9ÙA^()ResiduetheoremEvaluationofDefiniteIntegralsResiduetheoremSomeApplicationsofResidueTheoremResidueatInfinityµã3ê½nw·§)Û¼êÈ©¼ê3SÛ:k'©O)Û¼êÈ©§IOѼê3SÛ:?3ê¦f(z)3Û:b?3ê§Kþ`§Ò´¦f(z)3z=bSLaurentÐm¥(z−b)−1Xê34:¹e§±ÏLûO¦3êC.S.Wu1ù3ê½n9ÙA^()ResiduetheoremEvaluationofDefiniteIntegralsResiduetheoremSomeApplicationsofResidueTheoremResidueatInfinityµã3ê½nw·§)Û¼êÈ©¼ê3SÛ:k'©O)Û¼êÈ©§IOѼê3SÛ:?3ê¦f(z)3Û:b?3ê§Kþ`§Ò´¦f(z)3z=bSLaurentÐm¥(z−b)−1Xê34:¹e§±ÏLûO¦3êC.S.Wu1ù3ê½n9ÙA^()ResiduetheoremEvaluationofDefiniteIntegralsResiduetheoremSomeApplicationsofResidueTheoremResidueatInfinityµã3ê½nw·§)Û¼êÈ©¼ê3SÛ:k'©O)Û¼êÈ©§IOѼê3SÛ:?3ê¦f(z)3Û:b?3ê§Kþ`§Ò´¦f(z)3z=bSLaurentÐm¥(z−b)−1Xê34:¹e§±ÏLûO¦3êC.S.Wu1ù3ê½n9ÙA^()ResiduetheoremEvaluationofDefiniteIntegralsResiduetheoremSomeApplicationsofResidueTheoremResidueatInfinity3êOz=b:´f(z)m4:3b:Sf(z)=a−m(z−b)−m+···+a−1(z−b)−1+a0+a1(z−b)+a2(z−b)2+···üàÓ¦±(z−b)m(z−b)mf(z)=a−m+···+a−1(z−b)m−1+a0(z−b)m+a1(z−b)m+1+a2(z−b)m+2+···a−1(z−b)mf(z)Ðmª¥(z−b)m−1Xêa−1=1(m−1)!dm−1dzm−1(z−b)mf(z)z=bC.S.Wu1ù3ê½n9ÙA^()ResiduetheoremEvaluationofDefiniteIntegralsResiduetheoremSomeApplicationsofResidueTheoremResidueatInfinity3êOz=b:´f(z)m4:3b:Sf(z)=a−m(z−b)−m+···+a−1(z−b)−1+a0+a1(z−b)+a2(z−b)2+···üàÓ¦±(z−b)m(z−b)mf(z)=