复变函数1.3_handoutnew

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1、EC¼êØ$È1.3EC¼ê5)Û5o¡Mó(%°)êÆX2009c92F!:ê½Â)Û¼êC-R^NÚ¼ê~K8¹1!:2ê½Â3)Û¼ê4C-R^5NÚ¼ê6~Kziyilee@Dept.Math,HITEC¼êØ$ÈY2009c92F2/14!:ê½Â)Û¼êC-R^NÚ¼ê~K££½Â(4)é∀ε>0§∃δ>0§s.t.0<

2、z−z0

3、<δ§Òk

4、f(z)−w0

5、<ε§¡w0¼êf(z)z→z04§Plimf(z)=w0.z→z0limf(z)=w0⇔limu(x,y)=u0,limv(x,y)=v0.z→z0x→x0x→x0y→y0y→

6、y0½Â(ëY)XJ¼êf(z)3Nδ(z0)Sk½Â,limf(z)=f(z0)§z→z0¡f(z)3:z0?ëY.ziyilee@Dept.Math,HITEC¼êØ$ÈY2009c92F3/14!:ê½Â)Û¼êC-R^NÚ¼ê~K½Â(«SëY)ef(z)3«DSz:ÑëY§¡§3DSëY.5(k.4«þëY¼ê5)¼êf(z)3k.4«DþëY§Kf(z)(1)Uk.,(2)UÚ,(3)3k.4«DþëY.5¼êf(z)3k.4«DþëY§f:D→∆N§K∆´«§¼êz=f−1(w)3«∆þëY.ziyilee@Dept.Math,HITEC¼ê

7、Ø$ÈY2009c92F4/14!:ê½Â)Û¼êC-R^NÚ¼ê~K!:1!EC¼êê!¶2!)Û¼ê¶3!EC¼ê)Û¿^§i.e.ܨiù^¶4!äEC¼ê)Û{¶5!NÚ¼ê.ziyilee@Dept.Math,HITEC¼êØ$ÈY2009c92F5/14!:ê½Â)Û¼êC-R^NÚ¼ê~K½Â(ê)¼êf(z)3Nρ(z)Sk½Â§ef(z+∆z)−f(z)lim∆z∆z→03§K¡§´f(z)'uzê,Pf0(z).⇒ëY;½Â()e¼êf(z)3:z§Kf(z)3:z§df=udx−vdy+i(udy+vdx).

8、/ªþ§±wf(z)=u+iv©dz=dx+idy¦.⇐⇒ziyilee@Dept.Math,HITEC¼êØ$ÈY2009c92F6/14!:ê½Â)Û¼êC-R^NÚ¼ê~K½n()Û¼ê)«DSz:Ñü¼ê¡)))ÛÛÛ(KKK)¼¼¼êêê.51!(f(z)+ϕ(z))0=f0(z)+ϕ0(z);2!(f(z)ϕ(z))0=f0(z)ϕ(z)+f(z)ϕ0(z);f(z)0f0(z)ϕ(z)−f(z)ϕ0(z)3!()=2,ϕ(z)6=0;ϕ(z)ϕ(z)4!(f(ϕ(z)))0=f0(ϕ)ϕ0(z);5!f0(z)=1,(f−1(w))06=0.

9、(f−1(w))0ziyilee@Dept.Math,HITEC¼êØ$ÈY2009c92F7/14!:ê½Â)Û¼êC-R^NÚ¼ê~Kܨiù^∆f(z)lim∆z∆z→0f(z)=u(x,y)+iv(x,y)~µ¼êf(z)=z2,-z=x+iy,:f(z)=u+iv=z2=x2−y2+2xyi,u´u=x2−y2,v=2xy.dd:ux=2x,uy=−2y,vx=2y,vy=2x.uyµux=vy,uy=−vx.ziyilee@Dept.Math,HITEC¼êØ$ÈY2009c92F8/14!:ê½Â)Û¼êC-R^NÚ¼ê~K½n(ܨiù

10、^)e¼êf(z)=u(x,y)+iv(x,y)3«DSk½Â§Kf(z)3DS)Û¼ê¿^´µ(1)u(x,y)v(x,y)3DS¶(2)ux=vy,uy=−vx3DS¤á.(Øef(z)´)Û¼ê,Kf0(z)=u+iv=u−iu=v+iv=v−iu.xxxyyxyy½Â(ܨiù§)dܨiù^(½§µ®¼êuÚv¥Ò(½Ñ,§õ?¿~ê.ziyilee@Dept.Math,HITEC¼êØ$ÈY2009c92F9/14!:ê½Â)Û¼êC-R^NÚ¼ê~K½Â(NÚ¼ê)e¼êf(z)3«DS÷vLaplace§∂2f∂2f∂x2+∂y2=0§¡f(z

11、)NNNÚÚÚ¼¼¼êêê.ef(z)=u(x,y)+iv(x,y)´)Û¼ê§Ku(x,y)Úv(x,y)Ñ´NÚ¼ê.ziyilee@Dept.Math,HITEC¼êØ$ÈY2009c92F10/14!:ê½Â)Û¼êC-R^NÚ¼ê~K~1-12~1-13~1-14~1-16~~~1-12.y²¼êez=u(x,y)+iv(x,y)´)Û¼ê,¿¦Ñ§ê.)))ez=ex+iy=excosy+iexsiny§u(x,y)=excosy§v(x,y)=ex·siny(y²ez)Û§yÜ—iù^