复变函数1.5.1-3_handoutnew

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1、EC¼êØ$ȧ1.5)Û¼ê?êo¡Mó(%°)êÆX2009c97FE?ê?êV?ê8¹1E?ê2?ê3V?êziyilee@Dept.Math,HITEC¼êØ$ÈY2009c97F2/15E?ê?êV?ê£:1.Eê?êµÂñ!uÑ!ýéÂñ¶2.?ê§?êÂñ»9Âñ¦{§?êÂñSÜÄ5¶C½n§AíاмêVÐmª¶3.ÐEC¼ê½Â95"ziyilee@Dept.Math,HITEC¼êØ$ÈY2009c97F3/15E?ê?êV?êEê?ê5ÂñÂ

2、ñ5ÄVg5P∞P∞eak=αk+βki(k∈N)´E~ê§K?êak=(αk+βki)k=1k=1´äkEêê?ê§ak´Ï"PnPnPnPnSn=ak=(αk+βki)=αk+iβkk=1k=1k=1k=1¡?êÜ©Ú"P∞P∞P∞P∞rn=ak=(αk+βki)=αk+iβkk=n+1k=n+1k=n+1k=n+1¡?ê{"eÜ©ÚS43§=PnPnS=limSn=limαk+ilimβkn→∞n→∞n→∞k=1k=1K¡?ê´Âñ§S¡?êÚ"ÄK¡?êuÑ"ziyilee@Dept.

3、Math,HITEC¼êØ$ÈY2009c97F4/15E?ê?êV?êEê?ê5ÂñÂñ55P∞P∞P∞1©?êakÂñ§=?êαkÚβkÑÂñ"k=1k=1k=1â¢ê?êñÑ55ïÄEê?êñÑ5"P∞2.Âñ?êak{rnn→∞ªu0§=?ε>k=10§Ñ3êN§¦n≥N§k

4、rn

5、<ε.P∞3©?êÂñ7^µe?êakÂñ§K§Ïk=1akk→∞ªu0µlimak=0"k→∞ziyilee@Dept.Math,HITEC¼êØ$ÈY2009c97F5/15E?ê?ê

6、V?êEê?ê5ÂñÂñ5P∞4.?êÂñ¿©^µéu?êak§en→∞§k=1nP+makªu0§=?ε>0§Ñ3êN§¦n>k=nnP+mNm≥0§k

7、ak

8、<εk=n5©3?êc¡O½Kk§ØUC?êÂñ5"P∞P∞P∞e¢?ê

9、ak

10、Âñ§K?êakÂñ§¡akk=1k=1k=1ýéÂñ?ê"ýéÂñÂñ?ꧡ^Âñ?ê"ziyilee@Dept.Math,HITEC¼êØ$ÈY2009c97F6/15E?ê?êV?êEê?ê5ÂñÂñ5P∞P∞e?êak(ak=αk+

11、βki)ýéÂñ§K?êαkk=1k=1P∞ÚβkÑýéÂñ"k=1P∞?êuk(z)´ECþz¼ê§¡ù?ê¼êk=1?ê"P∞P∞eê?êuk(z0)Âñ§K¡z0:uk(z)Âñ:"k=1k=1XJ¼ê?ê3«DSz:Ñ´Âñ§K¡§3DSÂñ"¤kÂñ:8ܧ¡¼ê?êÂñ"ziyilee@Dept.Math,HITEC¼êØ$ÈY2009c97F7/15E?ê?êV?êEê?ê5ÂñÂñ5P∞3«DSÂñ¼ê?êuk(z)U(½ük=1¼ê§§3«DSz½:z0uéA¼ê

12、P∞?êÚµf(z0)=uk(z0)"ù(½¼êf(z)=k=1P∞uk(z)¡¼ê?êÚ"k=1ε−Nµ3?½:z∈Dþ§?ε>0§Ñ3PnêN=N(ε,z)§¦n≥N§k

13、f(z)−uk(z)

14、<ε.k=1XJN=6uεzÃ'§§=N=N(ε)§K¡?ê3«DþÂñu¼êf(z)"P∞Puk(z)⇒f(z)k=1ziyilee@Dept.Math,HITEC¼êØ$ÈY2009c97F8/15E?ê?êV?êEê?ê5ÂñÂñ5P∞eE?êuk(z)¤k3«DSÑ÷v^

15、uk(z)

16、

17、≤k=1P∞P∞ak§?êakÂñ§K?êuk(z)3«DS´k=1k=1Âñ"dA.dOK(`?êOK)Âñ?ê5µ1.3«DSÂñëY¼ê?ê§ÙÚE´DSëY¼ê"2.«DSÂñëY¼ê?ê÷DSz^Åã1wCÅÈ©¶=e¤kuk(z)§k∈N3DSëP∞Yuk(z)⇒f(z)§Kk=1RP∞P∞RRC(uk(z))dz=Cuk(z)dz=Cf(z)dzk=1k=1ziyilee@Dept.Math,HITEC¼êØ$ÈY2009c97F9/15E?ê?êV?êEê?ê5ÂñÂñ

18、5dA.d½ne?ê´«DS)ۼ꧿3DS?4«D0SÂñ§K?ê±Å¦?¿¼ê§=P∞euk(z)k∈N3«DS)Û§uk(z)⇒f(z)3D0⊂k=1DS¤á§Kf(z)3«DS)Û§nP∞P∞(n)d(u(z))=u(z)⇒f(n)(z)3D0⊂DS¤á"dznkkk=1k=1ziyilee@Dept.Math,HITEC¼êØ$ÈY2009c97F10/15E?ê?êV?