复变函数1.4_handoutnew

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1、EC¼êØ$È1.4EC¼êÈ©{!È©½Â9ÙÄ5o¡Mó(%°)êÆX2009c92Fý£EȩܽnؽȩÜúª?¿¼ê~K8¹1ý£2EÈ©3ܽn4Üúª5?¿¼ê6~Kziyilee@Dept.Math,HITEC¼êØ$ÈY2009c92F2/22ý£EȩܽnؽȩÜúª?¿¼ê~K!:1!EC¼êÈ©Vg!5!O¶2!ܽn!4´½n!ÜÈ©úª!pê¶3!)Û¼ê?¿5"ziyilee@Dept.Math,HITEC¼êØ$ÈY2009c92F3/22ý£EȩܽnؽȩÜúª?¿¼

2、ê~Kk½Â(1w)ec´{ü,x0(t),y0(t)∈C[α,β],000z(t)=x(t)+iy(t)6=0,K¡c111www.dk^1wÄ뤡ÅÅÅããã111www.ÃAÏ(²,±¤ïÄþ´1w½Åã1w.½Â()5½µ{ü4“_”§=÷X§$ħ¤¤«SÜo31?ý;éml,Ñå:Úª:=.ziyilee@Dept.Math,HITEC¼êØ$ÈY2009c92F4/22ý£EȩܽnؽȩÜúª?¿¼ê~KEÈ©½ne¼êf(z)=u(x,y)+iv(x,y)÷cëY§Kf(z)÷cÈ,RRRf(z)dz=udx−

3、vdy+iudy+vdx.ccc/ªþ,f(z)=u+iv©dz=dx+idy¦.5RRR1)Rc(f1(z)+f2R(z))dz=cfR1(z)dz+cf2(z)dz;2)Rcf(z)dz=c1fR(z)dz+c2f(z)dz,c=c1+c2;3)f(z)dz=−f(z)dz,c−;c−c4)XJ3cþ÷v

4、f(z)

5、≤M(M>0)§KR

6、f(z)dz

7、≤ML,L´cl.cziyilee@Dept.Math,HITEC¼êØ$ÈY2009c92F5/22ý£EȩܽnؽȩÜúª?¿¼ê~K½nÚíؽn(ܽn)Ref(z)3üëÏ«DS)Û§

8、Kf(z)dz´cÃ'.cRRRJ«:f(z)dz=udx−vdy+iudy+vdx.cccíØ(üëÏ«)e¼êf(z)3üëÏ«DS)Û§Kf(z)÷?Û^4È©0.ziyilee@Dept.Math,HITEC¼êØ$ÈY2009c92F6/22ý£EȩܽnؽȩÜúª?¿¼ê~K3õëÏ«S§Ü½nؤá.~µ¼êf(z)=131<

9、z

10、<2S)Û.z23ü±cþkz=eiϕ,dz=ieiϕdϕ,RdzR0ieiϕdϕR00c1z=πeiϕ=πidϕ=iϕ

11、π=−iπ,RdzR0ieiϕdϕR00Hc2z=R−πeiϕR=−πidϕ=iϕ

12、−π=iπ,dz=dz−dz=2πi.czc2zc1zziyilee@Dept.Math,HITEC¼êØ$ÈY2009c92F7/22ý£EȩܽnؽȩÜúª?¿¼ê~KíØ(õëÏ«)e¼êf(z)3õëÏ«DS)Û§4cuDSH¤üëÏ«D1§Kf(z)dz=0.c½n(ܽn)¼êf(z)3õëÏ«DS)Û§3DþëY§>.c=HΓ+γ−+γ−+···+γ−,Kf(z)dz=0.12ncziyilee@Dept.Math,HITEC¼êØ$ÈY2009c92F8/22ý£EȩܽnؽȩÜúª?¿¼ê~KÚî¨4ÙZ[úª½

13、Â(ؽȩ)Rzf(z)3üëÏ«DS)Û§KΦ(z)=f(ζ)dζ3DSz0)Û,Φ0(z)=f(z),¡Φ(z)f(z)ØØؽ½½ÈÈÈ©©©½¼¼¼êêê.½n(Úî¨4ÙZ[úª)eΦ(z)´f(z)3üëÏ«DS¼ê§KZzf(ζ)dζ=Φ(z)−Φ(z0),z,z0∈D.z0ziyilee@Dept.Math,HITEC¼êØ$ÈY2009c92F9/22ý£EȩܽnؽȩÜúª?¿¼ê~KÜúªÜúª½n(ÜÈ©úª)D>.´(½E)c§¼êf(z)3DS)Û§3DþëY§KÜÈ©úªµZ1f(ζ)dζf(z)=

14、,z∈D.2πicζ−zR1f(ζ)dζgKµÜÈ©,z/∈Dº2πicζ−z½Â(Ü.È©)L´?^4½4Åã1w§f(z)÷LëY§R1f(ζ)dζ¡Ü.È©.2πiLζ−zziyilee@Dept.Math,HITEC¼êØ$ÈY2009c92F10/22ý£EȩܽnؽȩÜúª?¿¼ê~K)Û¼ê?¿¼ê35½nR1ϕ(ζ)dÜ.È©F(z)=Ldζ(½¼êF(z)3?Û2πiζ−zعL:üëÏ«SÑk?¿¼ê§¿Z(n)n!f(ζ)dζf(z)=.2πic(ζ−z)n+1½n(?¿¼ê35½n)eü¼êf(

15、z)3«D