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WuChong-shi 1 §10.1 !G#$%C&’()*+#,-./0123456789:;bk,k=1,2,3,···,n=?f(z)@GA-BCD=@GEFG=H@CIJ5f(z)#:;=KICf(z)dz=2πinXk=1resf(bk).resf(bk)L&f(z)@bkM#N?=OPQf(z)@bk#RALaurentSTE(z−bk)−1#U?a(k)−12V10.1WXYZ[\]10.1=^_‘7:;bka./01γk=bγkc@GA=Hdefg=KhijFk!Cauchylmn?aLaurentSTo#U?pq=r5ICf(z)dz=nXk=1Iγkf(z)dz=2πinXk=1a(k)−1=2πinXk=1resf(bk).stuvwxyz={|}t~ }t ~ 2 {|}t~ = }t ~st2F f(z)@:;bM#N?= KI =r f(z)@z=b#RALaurentSTE(z−b)−1 #U?2F@ ;# = k ¡¢£⁄ N?2WuChong-shi§10.1 2 F¥ƒ§¤'“«b; f(z)#’‹ ;=K@b;#RA=f(z)=a−1(z−b)−1+a0+a1(z−b)+a2(z−b)2+···. (z−b)›STqfifl=(z−b)f(z)=a−1+a0(z−b)+a1(z−b)2+a2(z−b)3+···. a−1=limz→b(z−b)f(z).F–†‡·# f(z) ¶&P(z)/Q(z)=P(z)•Q(z)c@b;n‚RACD=b Q(z)#’‹„;=Q(b)=0=Q0(z)6=0=P(b)6=0=Ka−1=limz→b(z−b)f(z)=limz→b(z−b)P(z)Q(z)=P(b)Q0(b).”10.1 1z2+1@:;M#N?2»z=±i O#’‹ ;2resf(±i)=12z z=±i=∓i2.”10.2 eiaz−eibzz2@:;M#N?2»z=0 O#’‹ ;2resf(0)=limz→0z·eiaz−eibzz2.=limz→0eiaz−eibzz=i(a−b).F…ƒ§¤'“«z=b f(z)#m‹ ;=m≥2=f(z)=a−m(z−b)−m+a−m+1(z−b)−(m−1)+···+a−1(z−b)−1+a0+a1(z−b)+···.fifl›I(z−b)m=(z−b)mf(z)=a−m+a−m+1(z−b)+···+a−1(z−b)m−1+a0(z−b)m+a1(z−b)m+1+···.‰oa−1 (z−b)mf(z)#STqE(z−b)m−1 #U?= a−1=1(m−1)!dm−1dzm−1(z−b)mf(z) z=b.”10.3 1/(z2+1)3@:;M#N?2»z=±i O#¿‹ ;2resf(±i)=12!d2dz2(z∓i)3·1(z2+1)3 z=±i=12!d2dz21(z±i)3 z=±iWuChong-shi 3 =12!(−3)(−4)(z±i)−5 z=±i=∓316i.F `∞¤' ´Q∞;=lˆresf(∞)=12πiIC0f(z)dz,‰˜#C0 _∞;¯˘(˙r ¨o ˚˘)’¸#^ =@^ A4∞; ˝ f(z)#:;†˛:;2Fresf(∞)ˇe f(z)@∞RALaurentSTEz1 #U?2resf(∞)=12πiIC0f(z)dz=−12πiICf1tdtt2=−1t2f1t@t=0;RA— ?STEt−1 #U?=−f1t@t=0;RA— ?STEt1 #U?=−f(z)@z=∞;RA— ?STEz−1 #U?2F‰7 •56 Me M 1. =}tf(z) ∞ ~st= f(z) ∞ Æ tª z−1 ~ tŁØ−1=Œº 22. æ = z−1 ı f(z) ∞ Æ tª ~ łø =œß= ∞ f(z)~ =resf(∞) Ø 02 = ∞ f(z)~ = = Ø 02F ' ' 2 2 ?f(z)=1(z−1)(z−2)(z−3) ((q21(z−1)(z−2)(z−3)=Az−1+Bz−2+Cz−3.¿7 l‡?=A,B•C=¯ r ?f(z)@’‹ ;z=1,z=2•z=3;M#N?2 A=res1(z−1)(z−2)(z−3) z=1=12,WuChong-shi§10.1 4 B=res1(z−1)(z−2)(z−3) z=2=−1,C=res1(z−1)(z−2)(z−3) z=3=12.\ ?f(z) 5 ‹ ;=˙ Mm2 \=1(z−1)2(z−2)(z−3)=A(z−1)2+Bz−1+Cz−2+Dz−3. A=res1(z−1)(z−2)(z−3) z=1=12,B=res1(z−1)2(z−2)(z−3) z=1=34,C=res1(z−1)2(z−2)(z−3) z=2=−1,D=res1(z−1)2(z−2)(z−3) z=3=14.F ' 2N?lm^ !(#£⁄#&N?#£⁄=$%˝l!(•’lCD?#^ !(&U’(=r5 ˝)*,+ £⁄ ‰,l!(2WuChong-shi 5 §10.2- ./0 1235m¿4?#!(#5q I=Z2π0R(sinθ,cosθ)dθ,‚ER sinθ,cosθ#5m?=@!(!6I FG#2a78z=eiθ=Ksinθ=z2−12iz,cosθ=z2+12z,dθ=dziz,9:#!(;K7&z=I#-?@#@¸|z|=12Q =I=I|z|=1Rz2−12iz,z2+12zdziz=2πX|z|1res1zRz2−12iz,z2+12z.5m¿4?R(sinθ,cosθ)@!(!6[0,2π]IFG=rABC5m?Rz2−12iz,z2+12z@-?@#@¸I˛:;2”10.4£⁄!(I=Z2π011+εcosθdθ,|ε|12»DEI#˚FGH=IJ5I=Z2π011+εcosθdθ=I|z|=111+εz2+12zdziz=I|z|=12εz2+2z+εdzi=2πX|z|1res2εz2+2z+ε=2π·22εz+2 z=(−1+√1−ε2)/ε=2π√1−ε2.£⁄N?o=KL?2/(εz2+2z+ε)5fi7 ;=z=−1±√1−ε2ε,MNQOJ#›!&1= ’l$5’7 ;=z=(−1+√1−ε2)/ε=MQ-?@A2WuChong-shi§10.3OPQR 6 §10.3ST23˛U!(#lˆ&Z∞−∞f(x)dx=limR1→+∞R2→+∞ZR2−R1f(x)dx.5o‰V 6eW@=MlimR→+∞ZR−Rf(x)dxW@=L&!(XB=Y&v.p.Z∞−∞f(x)dx=limR→+∞ZR−Rf(x)dx.Z[=\‰fiV 6]W@o=OJ^l9P2@j=I =!(Z∞−∞f(x)dx _‘abcd#=ˇeefj7?#^ !(2•IJ a?f(x)gh&j?f(z)•&C˝ef^ !(ˇ:iN?lm£⁄=j^k (1)lIm\#!(;n5f./^ =£⁄If(z)dzo(2)@lI#;I#!(=pqr % £⁄#˛U!(st9u=pq ,-˚+ £⁄ (2vw[#xF\[ lI ;&@y=R&z#Iz@CR=ICf(z)dz=ZR−Rf(z)dz+ZCRf(z)dz.V10.2n{|R→∞2‰}=IJ+~%£⁄ZCRf(z)dz# 6B2$%f(z) m\# =‰ x #2e ?f(z) WuChong-shi 7 1.f(z)@Iz=4C56789:; MMCD#=@abIJ5:;o2.@0≤argz≤π ^A=\|z|→∞o=zf(z)’ Q0= ´Q #ε0=W@M(ε)0=b\|z|≥M=0≤argz≤πo=|zf(z)|ε2Œ 2 1 ~ = Ø stuv ICf(z)dz=ZR−Rf(z)dz+ZCRf(z)dz=2πiX resf(z). 2 =¡¢ £ ⁄¥ ~ƒ§ limx→±∞xf(x)=0~¤'“«=‹›=fifl v3.2=– limR→∞ZCRf(z)dz=0.† 6R→∞=r‡ Z∞−∞f(x)dx=2πiXIz=resf(z).”10.5£⁄l!(I=Z∞−∞dx(1+x2)32» oZ[·/I % # = I=Z∞−∞dx(1+x2)3=2πi·res1(1+z2)3 z=i=2πi·−3i16=38π.v{=&C´:iN?lm£⁄l!(#¶•‚„5’7)*”»#mC=e …‰j’ #¿ &C˝ :iN?lm£⁄˛U!(=IJ^k 1.lIm\#!(;n5f./^ =£⁄If(z)dzo2.@lI#;I#!(=pqr % £⁄#˛U!(st9u=pq ,-˚+ £⁄ (2WuChong-shi§10.3OPQR 8 ¶Q‰}#mC=r `´ˆ˜ ¯iN?lm£⁄l!(2F˘ f(x)˙}t=ł¨ Z∞0f(x)dx= Z∞0f(x)dx=12Z∞−∞f(x)dx, Ø˚' ظ 10.2~ = ˝˛ ~ˇ—= Z∞0f(x)dx=12Z∞−∞f(x)dx=πiX resf(z).F˘ Z∞0f(x)dx = }tf(z) ¨ = ˘f(z)=f(zeiθ), = ظ 10.3 ~ 2V10.3V10.4”10.6£⁄l!(Z∞0dx1+x42»NQ‰˜# !?f(x)=11+x4 x4#?= =IJ i]10.4#^ _¯abN0 R=_@ Ư b=…_¯ bNiRª ;2‰}=hiN?lm=5ICdz1+z4=ZR0dx1+x4+ZCRdz1+z4+Z0Ridy1+(iy)4=(1−i)ZR0dx1+x4+ZCRdz1+z4=2πires11+z4 z=eiπ/4=π21−i√2.† 6R→∞= &limz→∞z·11+z4=0, =hi m3.2=5limR→∞ZCRdz1+z4=0.WuChong-shi 9 Q r‡ Z∞0dx1+x4=√24π. Œ = '˚' ظ Ł~ 2Œ› }t1/(1+z4) z=eiπ/4Øz=ei3π/42 Œ 'º 2 Ø æ=˘ º u Z∞0dx1+x100,¸ π/50~ Ł = o ¸ Ł = ł 50 2 ı Œ ~ łø œ2˘ = Œ = Ł Ø Ł ıß ~ = = Œ = Ł ~ 2”10.7£⁄!(Z∞0dx1+x32»Z[=‰o: 4&2π/3# 5^ (]10.5)2V10.5ICdz1+z3=ZR0dx1+x3+ZCRdz1+z3+Z0Rei2π/3dx1+x3=1−ei2π/3ZR0dx1+x3+ZCRdz1+z3=2πires11+z3 z=eiπ/3=2π3e−iπ/6.† 6R→∞= &limz→∞z·11+z3=0, limR→∞ZCRdz1+z3=0.v{r‡ Z∞0dx1+x3=2π3e−iπ/61−ei2π/3=π3cosπ6=2π3√3.