第3章復(fù)變函數(shù)習(xí)題課

1、12 2zdz3i z :x013xix 30i 2zdz3i 0 012x2(3) i (3) i d x 133(3) i 6263i (3) i x 3i 3i 3 ,zx :x031C 2z dz 032xd x 93,zy i :y011C 2z dz 012(3)iy d(3)iy 3 263i 6263i , i3i i1C 2z dz 012()iyd()iy3i ,ziy :y01,zxi :x031C 2z dz 032()xi d()xi 69i 6263i 3;yx 10i 2(xdz1i z 2(x(1) i (1) i 1566i 31(3xz 2(x(12)xi

2、10 (1) i 1566i (1) i 10 23(2)xx i :x0121)2x i 31(3x41)2x i )ix 2)ix d xd xd x10102yx :x01xxi 2xix 01)iy 4C |zdzx1 1, 0y | 1z xy1 11C1:C,zx :x1 11C |z z dz11 |x x d x0 2:C2Cz ,i te:t0 2C |z z dz0 i te i te dit 0 diti 1C |z z dz2C |z z dzi zdz itedit5C dz1)2)1) |2,z 2itze 2itze dz |zzdzC 2ditiet20 20

3、2 dit4 i 2ite 2|2z z 4zC 2z4dz2C 1zdz2 i |4z z 16zC 4z16dz4C 1zdz:t02 2itedit2 4 i 2 i 4 8 i |zz|z2, |z4. 6| | 1z 224zz 35z dz0,224zz 0 z 1 3i 1 3i 2 1 224zz 35z | 1z 0.| | 1z coszzedz0,coszz 0 ,2 3,2 ,| 1z | 1z 0.7| | 1z 2(1)zez dz0,2(1)zez | 1z 0.1| |2z 2(1)z 1dz0,3(1)z 2(1)z 3(1)z z 12 32i 0 z 1,

4、 1|2z 1|2z 0.8|2| 1z 2z 2 i 21Ozeze22 e i dzC dzC 2z ze2z ze2z O1|zr 1 1z ( )14()z cosz 512i dzC 4!2 i 5(1)z cosz cosz 9|2z dzC 2()2z sinzO22z sinz2 2z 2 i sinz2 i cos2 0 () 221(1)(4)zz dzC |z1.5 1.5Oii 2i2i 1()3 211z 214z dzC dzC 1(6i 1zi 1zi 0 C dzC d ) z0 1(22)6iii 10| 1zi dzC 3(1)z 2371zz 1z dzC

5、 |za 2aaOa 1za za dzC za 1za 2 i 1za ia z a 22za 1a 11C dzC 3zcosz1C2C 1:C|z2 2:C|z3 dzC 3zcosz1C 2C 2C 3cosdzzz3cosdzzz2( )()0z coszdz1C 3zcoszcoszi ( cos ) z 0z i 0z 2!2 i dz2C 3zcoszi i () i 0 12O1| 1z a| 1a | 1,a aza (2)()z a zei ae| 1,a 5()zeza | 1z dzC 2!2 i 3()za zezedzC 2zsinzez |2zi 0z sinz

6、ez 2 i sinzez 0()z 2 i 13|(1)|2zi dzC 41z 32z O1i41z (1)z (1)z ()zi ()zi 41z 1 211z 1(2)211z 11z 11z 1(4)1zi 1iz 1(4i )32z 32z 32z 32z 32z dzC C dzC dzC dzC dz 142 i 32z 1()z 0 14i 2 i 32z ()z i 0 i 141()2 dzC 22zza 1za 1za aa dzC dzC 1(2 2 i )0 O,aa a,a0 i 1(2 2 i )2 i 2 i dzC 22za za ,aa ,aa 15( )

7、f z0z0z0()f z ( )g z( )f z( )g z z( )f z( )g z ( )f z0()g z ( )g z( )f z( )g z 0( )f zzz 0( )g zzz 0()f z0()g z 0( )f zzz dzC 12 i 0( )g zzz dzC 12 i dzC dzC 16z( )f zuiv varctanyx 0 x xv 1 221yx 2()yx y22xy 22xv 222()xy 2xyyv x22xy 22yv 222()xy 2xy22xv 22yv 0 v( )fz xiy 1z ( )f zlnzC xu xiv 22xy z

8、z zyv17uxe ( cossin )yyxy xy ( )f zu ,iv (0)fi xuxe ( cossin )yyxy xe sin y1 xxuxe ( cossin )yyxy 2xe sin yyuxe (cos ysinyy cos )xy 1 yyuxe ( 2sin y cosyy sin )xy xxuyyu 0, ( , )u x y( )fz xu xiv yiu xe cosyyxe sinxyxe sin y1 xie cos yxie sinyyxie cosxy i y zeix zei ze1 i iz zei ze1 i ( )f ziz ze(1

9、) i z ,C (0)f, i C , i( )f ziz ze(1 ) i z . i 18( , )u x y()xy ( )f z( )fz xu ( , )u x y3x 23xy xu23x 23y yu23x 6xy 23x 23y 23i x 6i xy 3 3i 2(x3 2()xiy 3i 2()xiy 3(1) i 2z( )f z(1) i 3zC xxu 6xyyu 6x xxuyyu 0, ( , )u x y23x y 3y 6xy 23y 6y 6y xiv yiu 6xy 23i y 2y 2)xyi 2(x2y 2)xyi u iv 22(4)xxyy 19( , )v x y y22xy ( )f z(2)fu ,iv 0 xv2xy 222()xy xxv 223(