北大数学物理方法(A)-复变函数教案12留数定理及其应用2

Outline1›ù3ê½n9ÙA^()ÔnÆêÆÔn{‘§|2007cSC.S.Wu1›ù3ê½n9ÙA^()OutlineùLJ:13ê½nOŽ½È©(Y)¹n¼êáȩ¢¶þkÛ:œ/õŠ¼êÈ©C.S.Wu1›ù3ê½n9ÙA^()EvaluationofDefiniteIntegrals(continued)ReferencesÇÂÁ§5êÆÔn{6§§7.4—7.6ù&œ§5êÆÔn{6§§4.2,4.3nÎ!X1Á§5êÆÔn{6§§5.3,5.4,5.5C.S.Wu1›ù3ê½n9ÙA^()EvaluationofDefiniteIntegrals(continued)IntegralsInvolvingTrigonometricFunction...IntegrandwithSingularityatRealAxisIntegralsInvolvingMultivaluedFunctionsùLJ:13ê½nOŽ½È©(Y)¹n¼êáȩ¢¶þkÛ:œ/õŠ¼êÈ©C.S.Wu1›ù3ê½n9ÙA^()EvaluationofDefiniteIntegrals(continued)IntegralsInvolvingTrigonometricFunction...IntegrandwithSingularityatRealAxisIntegralsInvolvingMultivaluedFunctionsý£µJordanÚn30≤argz≤π‰ŒS§|z|→∞žQ(z)⇒0§KlimR→∞ZCRQ(z)eipzdz=0Ù¥p0§CR´±:%!RŒ»þŒ=yz3CRþž§z=ReiθZCRQ(z)eipzdz=Zπ0Q(Reiθ)eipR(cosθ+isinθ)Reiθidθ≤Zπ0Q(Reiθ)e−pRsinθRdθC.S.Wu1›ù3ê½n9ÙA^()EvaluationofDefiniteIntegrals(continued)IntegralsInvolvingTrigonometricFunction...IntegrandwithSingularityatRealAxisIntegralsInvolvingMultivaluedFunctionsý£µJordanÚn30≤argz≤π‰ŒS§|z|→∞žQ(z)⇒0§KlimR→∞ZCRQ(z)eipzdz=0Ù¥p0§CR´±:%!RŒ»þŒ=yz3CRþž§z=ReiθZCRQ(z)eipzdz=Zπ0Q(Reiθ)eipR(cosθ+isinθ)Reiθidθ≤Zπ0Q(Reiθ)e−pRsinθRdθC.S.Wu1›ù3ê½n9ÙA^()EvaluationofDefiniteIntegrals(continued)IntegralsInvolvingTrigonometricFunction...IntegrandwithSingularityatRealAxisIntegralsInvolvingMultivaluedFunctionsý£µJordanÚn30≤argz≤π‰ŒS§|z|→∞žQ(z)⇒0§KlimR→∞ZCRQ(z)eipzdz=0Ù¥p0§CR´±:%!RŒ»þŒ=yz3CRþž§z=ReiθZCRQ(z)eipzdz=Zπ0Q(Reiθ)eipR(cosθ+isinθ)Reiθidθ≤Zπ0Q(Reiθ)e−pRsinθRdθC.S.Wu1›ù3ê½n9ÙA^()EvaluationofDefiniteIntegrals(continued)IntegralsInvolvingTrigonometricFunction...IntegrandwithSingularityatRealAxisIntegralsInvolvingMultivaluedFunctionsJordanÚn(‡:)30≤argz≤π‰ŒS§|z|→∞žQ(z)⇒0§KlimR→∞ZCRQ(z)eipzdz=0=yz3CRþž§z=ReiθZCRQ(z)eipzdz≤Zπ0Q(Reiθ)e−pRsinθRdθ≤εRZπ0e−pRsinθdθ=2εRZπ/20e−pRsinθdθy²'…3u°(OsinθŠC.S.Wu1›ù3ê½n9ÙA^()EvaluationofDefiniteIntegrals(continued)IntegralsInvolvingTrigonometricFunction...IntegrandwithSingularityatRealAxisIntegralsInvolvingMultivaluedFunctionsJordanÚn(‡:)30≤argz≤π‰ŒS§|z|→∞žQ(z)⇒0§KlimR→∞ZCRQ(z)eipzdz=0=yz3CRþž§z=ReiθZCRQ(z)eipzdz≤Zπ0Q(Reiθ)e−pRsinθRdθ≤εRZπ0e−pRsinθdθ=2εRZπ/20e−pRsinθdθy²'…3u°(OsinθŠC.S.Wu1›ù3ê½n9ÙA^()EvaluationofDefiniteIntegrals(continued)IntegralsInvolvingTrigonometricFunction...IntegrandwithSingularityatRealAxisIntegralsInvolvingMultivaluedFunctionsJordanÚn(‡:)30≤argz≤π‰ŒS§|z|→∞žQ(z)⇒0§KlimR→∞ZCRQ(z)eipzdz=0=yz3CRþž§z=ReiθZCRQ(z)eipzdz≤Zπ0Q(Reiθ)e−pRsinθRdθ≤εRZπ0e−pRsinθdθ=2εRZπ/20e−pRsinθdθy²'…3u°(OsinθŠC.S.Wu1›ù3ê½n9ÙA^()EvaluationofDefiniteIntegrals(continued)IntegralsInvolvingTrigonometricFunction...IntegrandwithSingularityatRealAxisIntegralsInvolvingMultivaluedFunctionsJordanÚn(‡:)30≤argz≤π‰ŒS§|z|→∞žQ(z)⇒0§KlimR→∞ZCRQ(z)eipzdz=0=yz3CRþž§z=ReiθZCRQ(z)eipzdz≤Zπ0Q(Reiθ)e−pRsinθRdθ≤εRZπ0e−pRsinθdθ=2εRZπ/20e−pRsinθdθy²'…3u°(OsinθŠC.S.Wu1›ù3ê½n9ÙA^()EvaluationofDefiniteIntegrals(continued)IntegralsInvolvingTrigonometricFunction...IntegrandwithSingularityatRealAxisIntegralsInvolvingMultivaluedFunctions0≤θ≤π/2žsinθ≥2θ/πZCRQ(z)eipzdz≤2εRZπ/20e−pRsinθdθ≤2εRZπ/20e−pR·2θ/πdθ=2εR·π2pR1−e−pR
=επp1−e−pR
→0∴limR→∞ZCRQ(z)eipzdz=0C.S.Wu1›ù3ê½n9ÙA^()EvaluationofDefiniteIntegrals(continued)IntegralsInvolvingTrigonometricFunction...IntegrandwithSingularityatRealAxisIntegralsInvolvingMultivaluedFunctions0≤θ≤π/2žsinθ≥2θ/πZCRQ(z)eipzdz≤2εRZπ/20e−pRsinθdθ≤2εRZπ/20e−pR·2θ/πdθ=2εR·π2pR1−e−pR
=επp1−e−pR
→0∴limR→∞ZCRQ(z)eipzdz=0C.S.Wu1›ù3ê½n9ÙA^()EvaluationofDefiniteIntegrals(continued)IntegralsInvolvingTrigonometricFunction...IntegrandwithSingularityatRealAxisIntegralsInvolvingMultivaluedFunctions0≤θ≤π/2žsinθ≥2θ/πZCRQ(z)eipzdz≤2εRZπ/20e−pRsinθdθ≤2εRZπ/20e−pR·2θ/πdθ=2εR·π2pR1−e−pR
=επp1−e−pR
→0∴limR→∞ZCRQ(z)eipzdz=0C.S.Wu1›ù3ê½n9ÙA^()EvaluationofDefiniteIntegrals(continued)IntegralsInvolvingTrigonometricFunction...IntegrandwithSingularityatRealAxisIntegralsInvolvingMultivaluedFunctions0≤θ≤π/2žsinθ≥2θ/πZCRQ(z)eipzdz≤2εRZπ/20e−pRsinθdθ≤2εRZπ/20e−pR·2θ/πdθ=2εR·π2pR1−e−pR
=επp1−e−pR
→0∴limR→∞ZCRQ(z)eipzdz=0C.S.Wu1›ù3ê½n9ÙA^()EvaluationofDefiniteIntegrals(continued)IntegralsInvolvingTrigonometricFunction...IntegrandwithSingularityatRealAxisIntegralsInvolvingMultivaluedFunctions0≤θ≤π/2žsinθ≥2θ/πZCRQ(z)eipzdz≤2εRZπ/20e−pRsinθdθ≤2εRZπ/20e−pR·2θ/πdθ=2εR·π2pR1−e−pR
=επp1−e−pR
→0∴limR→∞ZCRQ(z)eipzdz=0C.S.Wu1›ù3ê½n9ÙA^()EvaluationofDefiniteIntegrals(continued)IntegralsInvolvingTrigonometricFunction...IntegrandwithSingularityatRealAxisIntegralsInvolvingMultivaluedFunctions0≤θ≤π/2žsinθ≥2θ/πZCRQ(z)eipzdz≤2εRZπ/20e−pRsinθdθ≤2εRZπ/20e−pR·2θ/πdθ=2εR·π2pR1−e−pR
=επp1−e−pR
→0∴limR→∞ZCRQ(z)eipzdz=0C.S.Wu1›ù3ê½n9ÙA^()EvaluationofDefiniteIntegrals(continued)IntegralsInvolvingTrigonometricFunction...IntegrandwithSingularityatRealAxisIntegralsInvolvingMultivaluedFunctions¹n¼êáȩùaÈ©