复变函数1.5.6_handoutnew

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1、EC¼êØ$ȧ1.5)Û¼ê?êo¡Mó(%°)êÆX2009c915FÐEC¼ê~K8¹1ÐEC¼ê2~Kziyilee@Dept.Math,HITEC¼êØ$ÈY2009c915F2/22ÐEC¼ê~K)Ûmÿezez5n¼êV¼êLnzÐEC¼êe3¢¶«m[a,b]þ½ëY¼êf(x)§K3¹¢¶«m[a,b]E²¡«DS3)Û¼êF(z)§§3«m[a,b]þf(x)§r¼êF(z)¡f(x)3Eê«DS)Ûmÿ"ùp½Â)ÛmÿXJ3§7´"®¼êP∞xkP∞2k

2、+1P∞2kex=§sinx=(−1)kx§cosx=(−1)kxk!(2k+1)!(2k)!k=0k=0k=0Ù¥0=1§¢Cþx∈(−∞,+∞)§ECþ?êP∞zkP∞2k+1P∞2kez=§sinz=(−1)kz§cosz=(−1)kzk!(2k+1)!(2k)!k=0k=0k=03E²¡þÂñ§§Úþ´z¼ê"ziyilee@Dept.Math,HITEC¼êØ$ÈY2009c915F3/22ÐEC¼ê~K)Ûmÿezez5n¼êV¼êLnzP∞kk−1(x−1)lnx=(−1)§é∀

3、x∈(0,2]Âñ"lnx)Ûkk=1mÿµP∞kk−1(z−1)lnz=(−1)§3

4、z−1

5、<1S)Û"kk=1¼êlnzØ´¼ê§Ï½Â§?ê3AüSÂñ"5½nr)Û¼êVgí2¼êm'Xª§ïá¢C¼êm'Xª3E²¡þ)Ûmÿ"sin2z+cos2z=1§elnz=z§ez1·ez2=ez1+z2"¼êm©'Xª3E²¡þ)Ûmÿ("e,¢C¼ê´,©§)§¿3E²¡«DSk)Ûmÿ§K§ù)Ûmÿò´ù(©§3Ó«DS)Ûmÿ)©§)"ziyilee@

6、Dept.Math,HITEC¼êØ$ÈY2009c915F4/22ÐEC¼ê~K)Ûmÿezez5n¼êV¼êLnz¼êez½ÂµP∞zzke=§(?ê/VЪ)k!k=0ez=ex(cosy+isiny)§(z=x+iy)y²ù½Âd5§Äz=iϕ§kP∞keiϕ=ikϕk!k=0P∞2lP∞2l+1=(−1)lϕ+i(−1)lϕ(2l)!(2l+1)!l=0l=0=cosϕ+isinϕî.úª/ðªµeiϕ=cosϕ+isinϕz=x+iy§dez1+z2=ez1·ez2§ez=e

7、x+iy=ex·eiy=ex·(cosy+isiny)ziyilee@Dept.Math,HITEC¼êØ$ÈY2009c915F5/22ÐEC¼ê~K)Ûmÿezez5n¼êV¼êLnz5µ(1).z=x(y=0)§ez½Â¢ê¼ê½Â"z=iy(x=0)§eiy=cosy+isiny(2).{½n¤á§ez1+z2=ez1·ez2(3).

8、ez

9、=ex>0§argez=y¶3z²¡þez6=0ez=ex+iy=ex·eiy=ex·(cosy+isiny)ÌArgez=argez+2kπ.(4

10、).ez3z²¡þ)Û§(ez)0=ez(5).ez´±Ï2kπi±Ï¼ê§ez=ez+2kπiducosy=cos(y+2kπ)§siny=sin(y+2kπ)§ez=ex(cosy+isiny)=ex(cos(y+2kπ)+isin(y+2kπ))=ex·ei(y+2kπ)=e(x+iy)+i2kπ=ez+2kπiziyilee@Dept.Math,HITEC¼êØ$ÈY2009c915F6/22ÐEC¼ê~K)Ûmÿezez5n¼êV¼êLnz(6).4limezØ3§=e∞ÿÂ"z→∞lim

11、ez=∞§limez=0z=x→+∞z=x→−∞(7).Ïe−z·ez=e0=1§l−z1ez1z1−z2e=ez§ez2=e(8).Ø÷vÛ(Rolle)¥½n§3z²¡þ§ez=ez+2kπi§(ez)0=ez6=0"©¥½nؤá"(9).È©¥½nؤáR2πiϕR2πedϕ=(cosϕ+isinϕ)dϕ00R2πR2πiϕ=cosϕdϕ+isinϕdϕ=0§e6=000é?¿Eêz§keiz=cosz+isinzziyilee@Dept.Math,HITEC¼êØ$ÈY2009c915

12、F7/22ÐEC¼ê~K)Ûmÿezez5n¼êV¼êLnzdî.úªeiϕ=cosϕ+isinϕ§e−iϕ=cosϕ−isinϕsinz=1(eiz−e−iz)§cosz=1(eiz+e−iz)2i2P∞2k+1P∞2kqsinz=(−1)kz§cosz=(−1)kz(2k+1)!(2k)!k=0k=05µ(1).z=x(x¢ê)§sinz,cosz½ÂÏ~u{u¼ê½Â