1、= - 06 B. 1x GF+eW.*(x/n 1 / 37 N0N0N0N0 1. +X= , ? a+ib /j;G= / 4 3 5 1 3; ; (2 )(4 3 );7 1 1i ie i ii i i + + + + + . 7 i4 2 2 2 2e cos isin i i4 4 2 2 2 2 = + = + = 8 ( )( )( )( )3 5i 1 7i3 5i 16 13 i7i 1 1+7i 1 7i 25 25+ + = = + 9 ( )( )2 i 4 3i 8 3 4i 6i 5 10i+ + = + + = + : ( )3 1 i1 3 3 5= i
2、ii 1 i 2 2 2+ + = + 2.“r;G 4= ,.G D z z= G=1y? ( )2 2 22Rez w z z w w+ = + + ( )2 2 22Rez w z z w w = + ( )2 2 2 22z w z w z w+ + = + 5 *0 0Z1y?, G A ( )2 2 22Rez w z z w w+ = + + X:M1N,A“4A ;MA ( )2 2 22Rez w z z w w = + ( ( ) ( ) ( )( )22 2z w z w z w z w z wz z w w z w = = = +( )2 22Rez z w w= +
3、6|ET)4, 1y 4E, 7.6;G= /jj7 ?F9? 33 5 2 2; ; 1; 8(1 3 ); .cos sin7 1 9 9i i i ii+ + + + 7 ( )( )( )( )3 5i 1 7i3 5i7i 1 1 7i 1 7i+ + =+ + 38 16i 19 8i 17 e50 25 5i = = = 8 arctan19 = 8 eii = 2 = 2eii = 9 i i1 e e = = : ( ) 28 1 3i 16 3 + = = . ( )2 i38 1 3i 16 e + = ;32 2cos isin9 9 + (32 2cos isin 1
4、9 9 + = 3 2 2i .3 i9 32 2cos isin 1 e e9 9 + = = 8.A1 (1)i ,9!Qi (2)-1 ,9!Qi (3) 3 3i+,i . A i ,9!Qi ( )133 2 2 2 2i cos sin cos isin 0,1,22 2 3 3+ + = + = + = k ki k 1 3 1cos isin i6 6 2 2= + = +z 25 5 3 1cos isin i6 6 2 2= + = +z 3 9 9 3 1cos isin i6 6 2 2= + = z B -1 ,9!Qi ( ) ( )13 3 2 + 2 1 cos
5、 isin cos isin 0,1,23 3k k k+ = + = + = 1 1 3cos isin i3 3 2 2= + = +z 2 cos isin 1= + = z ,= - 06 B. 1x GF+e,W.*(x/n 3 / 37 3 5 5 1 3cos isin i3 3 2 2= + = z C 3 3i+ ,i i42 23 3i= 6 i 6 e2 2 + + = ( ) ( )1 12i4 4 2 2 4 43 3i 6 e 6 cos isin 0,12 2k kk + + + = = + = 1 1 i84 41 6 cos isin 6 e8 8 = + =
6、 z 91 1 i84 429 96 cos isin 6 e8 8 = + = z 9.A2e , 2i nz n= . A 11 0nz z + + + =L A(2ie nz = 1nz = 1 0nz = ( )( )11 1 0nz z z + + + =L ( n1 2 z. 1 6 = + : Im 0z aL z b = = , eib = .“r* L X a7 6 , ,u6?G 4?(2) ;1(3)1 | 2;(4)Re Im ;(5)Im 1 2.zz zz iz zz z= /jCO.E (2) |z-1|=|z|/j-$4 z= 12 (3) 1/j -i j 6
7、 1 2 j z, 6p4, 6), 4 Re(z)Imz /j-$4 y=x , #; zM = - 06 B. 1x GF+eW*(x/n 4 / 37 5Imz1D|z|/j 6-,0C NNNN 1. “rP41w zz= +; 6 | | 2z =,. Ai , iz x y w u v= + = +I 2 2 2 2 2 21 ii i i i( )ix y x yu v x y x y x yx y x y x y x y+ = + + = + + = + + + + + + j2 2 4x y+ =,p5 3 i4 4u iv x y+ = + p 54u x= ,34v y=
8、 + 5 34 4,u vx y= =p( ) ( )2 25 34 42u v+ =( ) ( )2 22 25 32 21u v+ =/j 6. 2. XP42w z=;GzM:, .P4jwM:,x .Aeiw =Fiw u v= + . 10 2,4r 06 B. 1x GF+eW*(x/n 5 / 37 1zt= ,I, 0z t . _22 201lim lim 01 1z ttz t = =+ + . (2) 0Re( )limzzz ; Az=x+yiIRe( )iz xz x y= +9 0 00Re( ) 1lim limi 1 iz xy kxz xz x kx k =
9、= =+ + nf = W. (1) 2 2( ) if z xy x y= +; = - 06 B. 1x GF+eW*(x/n 6 / 37 2 2( , ) , ( , )u x y xy v x y x y= =XM: . 2 2, 2 , 2 ,y u v vy xy xy xx y x y = = = = p? u vx y = u vy x = , 9z=0G A j( ) 0f z =p0u ux y = = ,0v vx y = = . pu,vjh _f(z)jh . (2) ( )f z. AA( ) if z u v= XD,I ( )u v u vx y x y =
10、= ( )u v vy x y = =+ ,u v u vx y y x = = 6 jRef(z)jh u=C1, 0u ux y = = jf(z)C-R0u ux y = = , u=C2 pf(z)jh . 5. |f(z)|=h . A j|f(z)|=C)CF |AA. 9C=0Iu=0,v=0,f(z)=0jh . 9C 0If(z) 0,v2( ) ( )f z f z C = u2+v2=C2 ITE)x,y6“r, 9 2 2 0, 2 2 0u v u vu v u vx x y y + = + = Y+XC-R44 ,44. 2 2 2 2 2( ) i 3 3 6 i
11、 3( 2 i) 3u vf z x y xy x y xy zx x = + = + = + = .(2) ( ) e ( cos sin ) ie ( cos sin )x xf z x y y y y y x y= + +. A ( , ) e ( cos sin ), ( , )=e ( cos sin )x xu x y x y y y v x y y y x y= +44 D e ( cos sin ) e (cos ) e ( cos sin cos )x x xu x y y y y x y y y yx = + = + e ( sin sin cos ) e ( sin si
12、n cos )x xu x y y y y x y y y yy = = e ( cos sin ) e (sin ) e ( cos sin sin )x x xv y y x y y y y x y yx = + + = + +e (cos ( sin ) cos ) e (cos sin cos )x xv y y y x y y y y x yy = + + = +pu vx y = , u vy x = pf(z)44 ,44. ( ) i e ( cos sin cos ) i(e ( cos sin sin )e cos ie sin (e cos ie sin ) i (e c
13、os ie sin )e e i e e (1 )x xx x x x x xz z z zu vf z x y y y y y y x y yx xy y x y y y y yx y z = + = + + + + = + + + + += + + = +10. A ( )( )3 3 3 32 2i , 0.0. 0.x y x y zf z x yz + + = + =“rA(1) f(z) Xz=04F5 (2)f(z) Xz=04%C?T,0; (3)f(0)= X A.(1)( ) ( )( ) ( )0 , 0,0lim ( ) lim , i ,z x yf z u x y
14、v x y = +606 B. 1x GF+eW*(x/n 8 / 37 (2)63L( )0( ) 0limzf z fz z “A f(z)=u(x,y)+iv(x,y) j f(z) X j D p u(x,y),v(x,y) X D D%C C-R 0; ,u v u vx y y x = = ( ) ( ) ( ) ( ) ( ), iv , , i ,f z u x y x y x y x y = = + ( ),u x yx x = ( ) ( ), ,u x y u x yy y y = = ( ),v x yx x = ( ) ( ), ,v x y v x yy y y =
15、+ = u (x,y),(x,y) X D1 D%C C-R G 4l (1) ( ) ( ) ( ) ( ) ( )1 i1 i ln 2 i 2 i1 i ln 1 i 1 i ln 1 i 4ln 2 i ln 2ln 2 2 44ln 2 2 42 41 i e e e e i ln 2i 2 4 4e e e cos ln 2 isin ln 24 4 2 e cos ln 2 isin ln 24 4kkkkk + + + + + + + + = = = + + += = + = + (2) ( ) ( ) ( )( )( ) ( )( )( ) ( )( )55 ln 3 5 l
16、n 35 ln3 i 2 i 5 ln 3 5i 2 5i5 ln 353 e ee ee cos 2 1 5 isin 2 1 53 cos 2 1 5 isin 2 1 5k kk kk k + + + + = = = + + += + + + (3)( )( )ii ln1 iln1 i ln1 i 0 2 ii 2 i 2 1 e e ee ekk k + + = = = = ( )( ) ( )1 i1 i 1 i1 i ln 1 i ln2 2 1 i ln1 i 2 i 1 i 2 i i4 4 i 2 2 i i 2 2 44 4 4 2 4 2 41 i e e2e ee e
17、 e e cos isin4 42(2)i2 24ek kkk k kkk+ + + + + + + + = = = = = = + = 18. A1;G 4l (1) ( )( ) ( )( )i 5i i 5i i 5 i 55 5 5 5 5 5e e e ecos 5i2 2e e 1 e e e e ch52 2 2+ + + + + = = + += = = =(2) ( )( ) ( )( ) ( )i 1 5i i 1 5i i 5 i 55 55 5 5 5e e e esin 1 5i2i 2ie cos1 isin1 e cos1 isin12ie e e esin1 i
18、 cos12 2 + = =+ =+ += (3)( ) ( )( )( ) ( )( ) ( ) ( )i 3 i i 3 ii 3 i i 3 i 2 2e esin 3 i sin 6 isin 22itan 3 icos 3 i e e 2 ch 1 sin 32i = = = + (4) ( )( ) ( )22 2i i2 2 2 22 2 2 2 2 22 21sin e e sin ch icos sh2isin ch cos shsin ch sh cos sin shsin shy x y xz x y x yx y x yx y y x x yx y + = = + =
19、+ = + + = +(5) ( ) ( )( )( ) ( )2arcsin i iln i 1 i iln 1 2i ln 2 1 i2 0, 1,i ln 2 1 i 2 k kk= + = + + = = + + L(6)( ) ( )( )i 1 i 1 2i i 2 1arctan 1 2i ln ln i2 1 i 1 2i 2 5 51 i arctan 2 ln 52 4k+ + + = = + + = + + 19. “r;G0; (1) sinz=2 ( ) ( )( )( )1arcsin 2 ln 2i 3i ln 2 3 ii1i ln 2 3 2 i212 iln
20、 2 3 , 0, 1,2zkk k = = = = + + = + + = L (2) e 1 3i 0z = e 1 3iz = + ( ) ln 1 3i ln 2 i 2 i31ln 2 2 i3z kk= + = + + = + + (3) ln i2z = ln i2z = i2e iz = = (4) ( )ln 1 i 0z + = ( ) 1ln 1 i ln 2 i 2 i ln 2 2 i4 4z k k + = + + = + + = - 06 B. 1x GF+eW*(x/n 10 / 37 20. 9z=x+iy“rA (1) sinz=sinxchy+icosxs
21、hy A ( ) ( )( )i i ii ii ie e e esin2i 2i1 . e e2isin ch icos .shx y x y iz zy x y xzx y x y+ + + = = = + (2)cosz=cosxchy-isinxshy A ( ) ( )( )( )( ) ( )( )i ii i i ii ie e 1cos e e2 21 e e21 e cos isin e . cos isin2e e e e.cos isin .2 2cos .ch isin .shz zx y x yy x y xy yy y y yzx x x xx xx y x y+
22、+ + += = += += + + + += = (3)|sinz|2=sin2x+sh2y A ( )i i1sin e e sin ch i cos sh2iy x y xz x y x y + = = + ( ) ( )2 2 2 2 22 2 2 2 2 22 2sin sin ch cos .shsin ch sh cos sin shsin shz x y x yx y y x x yx y= += + += + (4)|cosz|2=cos2x+sh2y Acos cos ch isin shz x y x y= ( ) ( )2 2 2 2 22 2 2 2 2 22 2co
23、s cos .ch sin .shcos ch sh cos sin .shcos shz x y x yx y y x x yx y= += + += + 21. AyG06,l,06DC wj|z|=1. (1) 5zCe dzz (2) 3cosCzdzz (3) 020tan 12,( ) 2Czdz zz z :- 2 2u x y= ,2 2xx y = +G-_B3 - ,v( )f z u i= +=_- A: = - 06 B. 1x GF+eW*(x/n 13 / 37 2u xx =2u yy =22 2ux = 22 2uy = 2 22 2 0u ux y + = ,
24、606 1 ( ) d2 ( )Cp z zi p z 1yC,p(z),MF *6(+| j,?06?): Af(z) XLDC FG jDDlim ( )zf z A= I ( ) , ,1 ( ) d, .2 Cf z A z DfA z Gi z + = GjCp $G j. A XD+ 0G,= k4 _ Vf,_5 )fF_闭 ( ) ( ) 0s z s z + = u闭9 ( ) cos sins z A z B z= + A, B 以. 200(0) ( 1) 1 1(2 )!nnznzA An= = = =+a分 2 101(0) sin cos ( 1) 0 0(2 1)
25、!nnznzs z B z Bn= = + = = = p以 20( 1) cos .(2 )!nnnz z Rn= = =+ 令令.A4 0nnC= f分0nnnC z= ,f z以j 令 A j4 0nnC= f A 11lim .nnnnnC Z zC Z += 9以 0nnnC z= ,f z以j 令 I1z= )+X A#A 1 = 9以 0 1 闭91lim 1nn nCC += 闭M5闭G 以0以? 2 5 4 31 1 1 1 , 1 1( 1) ( 1) ( 1) ( 1) zz z z z z= + + L. 5A以 1z = | ( )f z ,闭3W,w叶F TZ5 闭
26、(70Z|!.,jx? 代闭 =), z称令| f(z),4闭叶,=|闭3W,w叶.5p5 5闭G 以0以?=| X 0 1 1z06 6B. 1x G化+e,t.*(x/n 22 / 37 (令)ctan dz z n j 化j| z|称n A 代c csin tan d dcos zz z zz= j X 化 tan z 闭9 12kz k= + (k称代,令,以( n-令)04闭G06 (令) 0cos d5 4cosm - 0 - j , - p 以1 cos d2 5 4cosmI = 令 11 sin d2 5 4cosmI = I闭9 i1 1 ei d2 5 4cosmI I+
27、 = A iez = d 1 di zz = 2os 12c z z += I ( )1 21211 di2 i15 421 d2i 5 2 1mzmzz zI Izzzz zz=+ = + = + 0- ( ) ( )25 2 1mzf z z z= + X| z|称令 闭90Z1 闭G0积6 (令) ( )20sin 2 d1x xx x+ 成代令 ( ) ( )211R z z z= + f R(z) X.b:9.0,w叶 z称代以 叶j 6以 r j z以,: z 6 化r 叶 R,与-R, -r成, 叶r,与r, R成5成1i“4叶i“4 790Z,w叶 i, _ ( ) ( ) (
28、 )222i 201 e 1 eIm d Im 2i Res ,i lim d2 2 11 rrx izcI x R z zz zx x+ = = + 6代, 代积a积令 成代 X -A 4 z称化+iy (- 积 y 积+ ) : 令( )ln2 2ez z aaf zz z= = ( )ln2 2ei c af c yc y+ = + ( ) ln2 2ei d dc af c y y yc y+ + + = + f 5p 以 0积 6( )iidccf z z+_. X,令a ( ) ( ) ( )i ii id lim d lim dc cc c ABR R RRf z z f z z
29、 f z z+ + + += = z AB j,=令M 化-iR 化+iR ,4 630积6 6B. 1x 北化+e,W.*(x/n 24 / 37 (以) .y kx= (k j. ) 代 2 2 2 21 ix yw z x y x y= = + + 2 2 2 2 2 2x y kxu vx y x y x y= = =+ + + v ku= u 1w z= 6 y kx= m5-$4 v ku= . 以. ;北 j+, u67.Im4;m5x 令 Im( ) 0, (1 i)z w z = + 代 (1 i) ( i ) ( ) i( + )w x y x y x y= + + = +
30、 , . 2 0.u x y v x y u v y= = + = . u (1 i)w z= + 6 Im( ) 0,z m5 Im( ) Re( )w w . (以) Re(z)代. 代积Im(z)积令, iw z= . 代A z称x+iy, x代, 代积y积令. 2 2 2 2 2 2i i i( i ) ix y y xwz x iy x y x y x y= = = = + + + + Re(w)代. Im(w)代. 9以 w称u+iv, I 2 2 2 2,u vy xu v u v= =+ + -j 代积 y积令,I 2 22 2 1 10 1,( )2 2u u vu v+ u
31、 iw z= 6 Re( z)代, 代积Im(z)积令.mj Re(w)代,Im(w)代, 1212w (以12 ,代j 6以12 j z以I 6) 3. “r w称z以 u z称i,4IhRv) bi w称z以64b9hRv) bI= 3Wm4 w称z以 u z 令M:09hRv) b= 3Wm4 w称z以 u z称代 ,4I jMG j+,mjx .以. (令) -$4R Re( z)称叶令,Im(z)称叶以问 (以) 令V以 j+, Im( ) ,0 2z 代代积 y积 , 0 2 .f i=e e e ( 0,0 ) e 1,0 argz x y xw x y w = 代,代积Im(z
32、)积 , 0 2 9mj |w|令, 0 argw 令 以9m4j令Mi4-令积Re( w)积令,Im(w)称代 I9m4. 成代x+X9m4 1 1w z= 6| z|令 9mj| w令|积令,+X6以?43t9m4. 1211i1www+= 6| w令|积令 9mj: z令M Im( w以)代, f 与+X令r-8 23 2w w= 9mj9Ij.EI令M0 与+X6以?43t9m4 3311www=+ 6 j+,9mj9I与-令,令成I令M. 8u22 112 113 22 22 13 2 11111 1i 1111 1 1 1( )1 1 21 1i 1 11 1zzzzwww ww
33、zw w zww + = = = = = + + + + + + . 令8. “r*6 CO.E Re( ) 0z代I9m4. 成代+X 1 ezw = 6 j+,9mj9I(代,令)I # z令MRe(w令)代+X 1211ln1www+= 6 z令M9mj9I(-,-令成I 6,F+, +X 3 2iw w= 6 j+,9mj : z 6GI: z令M+X 4 3lnw w= 6j+,9mj z令V以 代积Im( w4)积 ,Re(w4)代0 与+X42 iw w= 9mjp“r j+,8u e 1lne 1zzw+= . 令9. “r6Im( z)积令 6| z|积令 以9m4j: z令
34、M Im( w)代 I9m4. 成代+. 以代. 9m 4 cosw z= 6 z 令V 以 j +,代积Re(z)积,Im(z)代 以9m4j 令M:Ix j+,. 成代 -j 1cos ( )2 iz izw z e e= = + 积以6成代j w令称iz 12 eww = 3 221 1( )2w w w= + +a cosw z= XpR j+, 叶成代p以 令 w令称iz 6 z 令V +, 8 E 2 9m j代积Im(w令)积,Re(w令)积代. 以 12 eww = 6 j+,9mj 6I: z 6G|w以|积令,Im(w以)代. 3 221 1( )2w w w= + 6 j
35、+,9mj; z令MIm(w)积代. N3N3N3N3 令.A9与, f(t)%.CG 叶 I叶以 f(t)j,w-8 9叶f9 + = 0 dsin)()( tbtf z ( )+ =0tdtsin2)( tfb 以 f(t) j - 8 9叶 f 9+ = 0 cos)()( tdwatf = - 06 B. 1x GF+et.*(x/n 28 / 37 + =02 tdtcf(t)( osa A j dGtf ti+= e)(21)( )(G j f(t),G ) 6.“r1V -8 1, 0sgn 1, 0| | ttt tt ,G * sgn( ) ( ) ( ).t u t u t= 8u 1 1Fsgn( ) F( ( ) F( ( ) ( ) ( ) ( )2 2 ( ) ( )t u t u t i ii i = = + +
