复变函数讲义2

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1、12ÙECê¼ê2.1ECê¼ê¥‰ŒêÆÆ2013c92.1ECê¼ê(¥‰ŒêÆÆ)12ÙECê¼ê2013c91/7½ÂDenition(½Â1)E´E²¡þ˜‡:8,eéuE¥z˜‡z,U˜½5Æk˜‡Eêw†ƒéA,K¡3Eþ½Â˜‡EüŠ¼ê,PŠw=f(z)(z2E);XJéugCþz˜‡Š,U5ƆƒéAwØŽ˜‡,K¡w=f(z)´õŠ¼ê.2.1ECê¼ê(¥‰ŒêÆÆ)12ÙECê¼ê2013c92/7½ÂDenition(½Â1)E´E²¡þ˜‡:8,eéuE¥z˜‡z,U˜½5Æk˜‡Eêw†ƒéA,K¡3Eþ½Â˜‡EüŠ¼ê,

2、PŠw=f(z)(z2E);XJéugCþz˜‡Š,U5ƆƒéAwØŽ˜‡,K¡w=f(z)´õŠ¼ê.Example•Ñe=¼ê´üŠ¼ê,=´õŠ¼ê?pw=jzj,w=z2,w=nz(n2N,…n2),w=Argz(z6=0),f(z)=1(z6=0).z2.1ECê¼ê(¥‰ŒêÆÆ)12ÙECê¼ê2013c92/7½ÂDenition(½Â1)E´E²¡þ˜‡:8,eéuE¥z˜‡z,U˜½5Æk˜‡Eêw†ƒéA,K¡3Eþ½Â˜‡EüŠ¼ê,PŠw=f(z)(z2E);XJéugCþz˜‡Š,U5ƆƒéAwØŽ˜‡,K¡w=f(z)´õŠ¼ê

3、.Example•Ñe=¼ê´üŠ¼ê,=´õŠ¼ê?pw=jzj,w=z2,w=nz(n2N,…n2),w=Argz(z6=0),f(z)=1(z6=0).z)üŠ¼ê:w=jzj,w=z2,f(z)=1(z6=0);pzõŠ¼ê:w=nz(n2N,…n2),w=Argz(z6=0).2.1ECê¼ê(¥‰ŒêÆÆ)12ÙECê¼ê2013c92/7Denition(½Â2)w=f(z)´8ÜEþüŠ¼ê,XJéuE¥?¿ü‡ØÓ:z19z2,§‚3¼êŠ8ÜE0¥éA:w1=f(z1)9w2=f(z2)•ØÓ,K¡w=f(z)´8E¥˜‡˜˜Nì(½V•üŠN

4、ì).½ö`,w=f(z)V•üŠ/r8EN¤8E0.2.1ECê¼ê(¥‰ŒêÆÆ)12ÙECê¼ê2013c93/7Denition(½Â2)w=f(z)´8ÜEþüŠ¼ê,XJéuE¥?¿ü‡ØÓ:z19z2,§‚3¼êŠ8ÜE0¥éA:w1=f(z1)9w2=f(z2)•ØÓ,K¡w=f(z)´8E¥˜‡˜˜Nì(½V•üŠNì).½ö`,w=f(z)V•üŠ/r8EN¤8E0.~w=z2.:w=1kü‡”z=1Úz=1.2.1ECê¼ê(¥‰ŒêÆÆ)12ÙECê¼ê2013c93/7Denition(½Â2)w=f(z)´8ÜEþüŠ¼ê,XJé

5、uE¥?¿ü‡ØÓ:z19z2,§‚3¼êŠ8ÜE0¥éA:w1=f(z1)9w2=f(z2)•ØÓ,K¡w=f(z)´8E¥˜‡˜˜Nì(½V•üŠNì).½ö`,w=f(z)V•üŠ/r8EN¤8E0.~w=z2.:w=1kü‡”z=1Úz=1.Example(~1)w=az,a6=0.2.1ECê¼ê(¥‰ŒêÆÆ)12ÙECê¼ê2013c93/7Denition(½Â2)w=f(z)´8ÜEþüŠ¼ê,XJéuE¥?¿ü‡ØÓ:z19z2,§‚3¼êŠ8ÜE0¥éA:w1=f(z1)9w2=f(z2)•ØÓ,K¡w=f(z)´8E¥˜‡˜˜Nì(½V

6、•üŠNì).½ö`,w=f(z)V•üŠ/r8EN¤8E0.~w=z2.:w=1kü‡”z=1Úz=1.Example(~1)w=az,a6=0.V•üŠ/r4z²¡N•4w²¡.2.1ECê¼ê(¥‰ŒêÆÆ)12ÙECê¼ê2013c93/7Denition(½Â2)w=f(z)´8ÜEþüŠ¼ê,XJéuE¥?¿ü‡ØÓ:z19z2,§‚3¼êŠ8ÜE0¥éA:w1=f(z1)9w2=f(z2)•ØÓ,K¡w=f(z)´8E¥˜‡˜˜Nì(½V•üŠNì).½ö`,w=f(z)V•üŠ/r8EN¤8E0.~w=z2.:w=1kü‡”z=1Úz=1.Exa

7、mple(~1)w=az,a6=0.V•üŠ/r4z²¡N•4w²¡.
ea=r(cosq+isinq),Kw=reiqz,=•þwd•þz²˜‡^=C†Ú˜‡ƒqC†.2.1ECê¼ê(¥‰ŒêÆÆ)12ÙECê¼ê2013c93/7Problem(2.1.1)3C†w=ize§±jz1j=1C¤Ÿo•‚º2.1ECê¼ê(¥‰ŒêÆÆ)12ÙECê¼ê2013c94/7Problem(2.1.1)3C†w=ize§±jz1j=1C¤Ÿo