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!# $%&’()*+,-./0(201)1.2343w=ln(1+z)56743 89:5;(1)argz$=?(2)arg(1+z)$=?(3)z$@=ABC?(4)1+z$@=ABCD2.E43f(z)FGHIJKGLMN COGLP1QRSTU VWOAXB YZ1ZCf(z)dz(1)[\]^_‘% a[bcdef%?(2)[\]^_f% a[bcde‘%?(3)[\]^_)bcde ‘%?(4)[\]^_)bcde f%D3.g43f(z)Fz=aWMN f(a)=f0(a)=···=f(n−1)(a)=0,f(n)(a)6=0,Y43f0(z)/f(z)Fz=aWPh3O(1)1−n?(2)n−1?(3)−n?(4)nD4.z=∞5f(z)=1sinzP(1) ijc?(2)klc?(3)mnc?(4)opqlcD5.Γ(z)Γ(1−z)=πsinπzPrsJKO;(1)tuv?(2)wxyz0Rez1?(3){|uvRez0?(4)}|uvRez1D~-(101) MN43f(z)F P37O 3 v(x,y)=xx2+y2 f(z)D2 -(201) 43lnz−1z+1Fz=∞P KL O 3 lnz−1z+1 z=∞=0D -(401) Z1;(1)Z∞−∞dx(x2+1)(x2−2xcosθ+1),0θπ, θ6=π/2.(2)Z∞−∞sinxx(x2+4)dx. -(101) f(t)=1πZπ0cos(tcosθ)dθ 8 P 43F(p)D ¡¢£⁄¥ƒ§¤,-(201)1.(2)(41)2.(4)(41)3.(4)(41)4.(4)(41)5.(1)(41)~-(101)'“1 «u(x,y)P‹›1du(x,y)=∂u∂xdx+∂u∂ydy=∂v∂ydx−∂v∂xdy=−2xy(x2+y2)2dx+x2−y2(x2+y2)2dy,Z1 u(x,y)=yx2+y2+C. f(z)F P37O 3 fifl y=0– u(x,y)=0,†‡· «C=0.¶• f(z)=y+ixx2+y2=iz∗zz∗=iz.∂u∂x(21)∂u∂y(21) «u(x,y)(21)Z1‚3C=0(21)„” «f(z)(21)'“2»…„”‰«f(z):v(x,y)=xx2+y2=12z+z∗zz∗=121z∗+1z4=12iiz∗+iz=12iiz−iz∗=f(z)−f∗(z)2i†‡· »… ¿f(z)=iz.( 5` ´`21) -(201)lnz−1z+1=ln1−1z1+1z=∞Xn=1(−)n−1n−1zn−1zn=∞Xn=1(−)n−1n[(−)n−1]1zn=−2∞Xn=012n+11z2n+1,|z|1. ˆ˜«¯U(31)˘71˙¨ (31)ln1−1z (41)ln1+1z (41)„”(41)˚¸ ˝(21) -(1)(201)˛˝ˇO — G Z1I1(z2+1)(z2−2zcosθ+1)dz. h3 I1(z2+1)(z2−2zcosθ+1)dz=ZR−R1(x2+1)(x2−2xcosθ+1)dx+ZCRI1(z2+1)(z2−2zcosθ+1)dz=2πiX res1(z2+1)(z2−2zcosθ+1)5=2πi1(z2+1)(z2−2zcosθ+1)Fz=iPh3+1(z2+1)(z2−2zcosθ+1)Fz=eiθPh3. • Z43Fz=iPh3=12i(−2icosθ)=14cosθ, Z43Fz=eiθPh3=1(ei2θ+1)2isinθ=12cosθ(cosθ+isinθ)2isinθ=−icosθ−isinθ4cosθsinθ=−i14sinθ−14cosθ. – :Olimz→∞z·1(z2+1)(z2−2zcosθ+1)=0, limR→∞1(z2+1)(z2−2zcosθ+1)dz=0.Æ ª ˛ R→∞ ¿Z∞−∞1(x2+1)(x2−2xcosθ+1)dx=π2sinθ.Z1˝ˇ(21) Z43(21)h3 (31)ŁWiØh3 (41)ŁWeiθØh3 (41) (21)ª (31) -(2)(201)˛˝ˇŒº8.7 G Z1Ieizz(z2+4)dz. h3 Ieizz(z2+4)dz=Z−δ−Reixx(x2+4)dx+ZCδeizz(z2+4)dz=ZRδeixx(x2+4)dx+ZCReizz(z2+4)dz6=2πie−2−8=−πi4e−2.:Olimz→∞1z(z2+4)=0, Jordan limR→∞ZCReizz(z2+4)dz=0. :Olimz→0z·eizz(z2+4)=14, limδ→0ZCδeizz(z2+4)dz=−πi4.Æ ª ˛ R→∞,δ→0 ¿Z∞−∞eixx(x2+4)dx=πi4 1−e−2. ¿ æª Z∞−∞sinxx(x2+4)dx=π4 1−e−2.Z1˝ˇ(21) Z43(21)h3 (31)ŁWz=2iØh3 (41) (41) Jordan (21)ª (31) -(101)F(p)=1πZπ0pp2+cos2θdθ=12πZ2π0pp2+cos2θdθ=12πI|z|=1pp2+14(z+z−1)2dziz=4p2πiI|z|=1zz4+2(2p2+1)z2+1dz.˜ ζ=z2 z P|z|=1, Oζ P|ζ|=1ı ¶•F(p)=4p2πiI|ζ|=11ζ2+2(2p2+1)ζ+1dζ=4p×1ζ2+2(2p2+1)ζ+1F˘ LPh3X.71ζ2+2(2p2+1)ζ+1PŁW5ζ=−2(2p2+1)±p4(2p2+1)2−42=−(2p2+1)±2ppp2+1,¶•1ζ2+2(2p2+1)ζ+1F˘ L ,łŁW ζ=−(2p2+1)+2ppp2+1,ø 5,œ W h3O12ζ+2(2p2+1) ζ=−(2p2+1)+2p√p2+1=14ppp2+1. æ ¿F(p)=1pp2+1.cosωtPLaplace (21) Oߢ |z|=1PZ1(21)˜ ζ=z2(11)ŁWØh3 (21)ª (21)rs ;(Rep0)(11)8