1、- 1 - 5 s 1./ L9 s+idzz302b 11 i3+ L21 L 3 3 wL ( ) ( ) = 0ImRe dzzfdzzfCC T T b A b 7 () zzf = 1: =zC 5 ( )zf x 3+i C2C1O C3i C4y (z) 3 - 2 - () =20ReReiiCdeedzzf ()=+=200icosisincos d () =20iiImIm deedzzfC()=+=200cosisinsin d 4 1zz= # Os T 2iCzdz =v C_ |1z = b 12iCCzdz dzz=vv Os T 59 s dzzzC C_ 1
2、2=z 2 4=z 1y 2| =z 2| =z 4|2= zzz V7zz4= # i422|2|2|4=dzzdzdzzzzCzZ2y C 4| =z 16|2= zzz V7zz16= # i844|4|4|16=dzzdzdzzzzCzZ6 4E/ sb O Os Tb 7 wL_9 / sb 1Czdzze2 1|2:| =zC 222Cdzzav :| |Cza a = 3i21zCedzz +v :| 2i | 3/ 2Cz = 43Czdzz v :| | 2Cz= 523,(1)(1)Cdzzzv:| | 1Cz r= wL 722(1)(4)Cdzzz+v :| | 3/
3、2Cz= 8sinCzdzzv 1|:| =zC 9Cdzzz22sin 2|:| =zC 105zCedzzv 1|:| =zC 1 Cauchys T=Czzzeedzzei2i2222 2 1=+=+=CCazaazdzazazazdzi1i2122 2 +=Cdzazdzazaazdz 112122i0i221aa = 3 Cauchys Tii i2i/( i)2i /1-zz zCC zedz edz z eezzz =+=+vv- 3 - 4 5 6 O T ( 0 7y$f i=z C = i2=z C # # Cauchys T =+=+=+31|i|2231|i|2222)
4、4)(1()4)(1()4)(1(zCzzzdzzzdzzzdz=()() ()()=+=+31|i|231|i|2i4i1i4i1zzdzzzzdzzzzi2i2)4i)(1i2)4i)(1i2=+=zzzzzz 033=8 Cauchys T0sin2isin | 0zCzdzzz= =v9 p T () 0sini22sin22=zCzdzzz10 p T(4)052i i() |4! 12zzzCedzez =v89 / 5 13i2izedz 20i6ch 3zdz 3i2-isin zdz 410sinzzdz 5i0(i)zzedz 6i211tan(1i )coszdzz+L
5、b 13i23i2ii02zzeedz=“ 200i/6i61ch 3 sh 3 | i/33zdz z = =3ii2i-i-i -i1cos2 sin2 1sin ( ) | ( sh 2 )i224 2zz zzdz dz =41100sin (sin cos ) | sin1 cos1zzdz zz z= =5ii00( i) (i1 ) | 1cos1i(sin11)zzzedz ze= =+6i2i 2 21211tan 1 1(tan tan / 2) | (tan1 tan 1 th 1) i th1cos 2 2zdz z zz+=+ =+ + +99 / s 143(),:
6、|412iCdz C zzz+=+v _ 222i,:|1|61Cdz C zz=+v _ 312123cos, :|2 :|3CC Czdz C z C zz=+=v _ _ - 4 - 416,i25CdzCziv _ +53,|1 :|1()zCedz a a C zza=v _ 143()2i(43)14i12iCdzzz +=+v“ 22| i|1 | i|12i 2 /( i) 2 /( i)01- iCz ziz izdz dz dzzzz= +=+=+= vv v312 1 200333cos cos cos 2 i 2 i(cos ) | (cos ) | 02! 2!zzC
7、C C C Czzdz dz dz z zz =+= =vvv4 2iiCdzz=v5 |1a H331/( ) | | 1 0()zCeza z dzzav # “ |1a wL H210Cdzz=vb wL C = H0212i(1)| 0zCdzz= =v wL C H21/ z C =#210Cdzz=vb 11/ s M$s 2 ? M V 1$ I $ 1| 2zzdzz=v 2| 4zzdzz=v 2i| 2 02i 0zzdz e dz=v 2i| 4 04i 0zzdz e dz= =v# sMb ? M V 1yzz Bf b 12 ! u D z D = |1z = iB
8、 D = iBH wL C z 201Re .14zd=+ f 211 + #9 V 0 z iBH wL Cs Hs 1b5i1222000 011 2icos.14 2zeddxd dxe =+ =+ + s0s 2i1 e+ - 5 - B1 C1 C2 B2 MNE FB GH# 201Re .14zd=+13 !1C 2C M M Na e wL usY1B 2B b1B 2B s Bb T ()f z 1B B 2B B =1C a2C 9 12() ()CCf zdz f zdz=vvb 1B B ()f z f 5 O () 0MENGMfzdz=v () 0MHNFMfzdz
9、=v5 () () () ()NGM MEN MHN NFMf zdz f zdz f zdz f zdz+=+12() ()CCf zdz f zdz=v vb 14 ! C V a -a _e wL a , k a -a C 9 sCdzazz22b i a C =7 -a C H =+=+=CazCazzdzazazzdzazzii222 b ii a C =7 a C H =+=ccazazzdzazazzdzazzii222 iii a a C = H !1C ,2C sY aa , sl P21,CC ( C = OM9c5 # Cauchys T =+=+=ccciiidzaza
10、zzdzazazzdzazz21222 iv a a C H Cauchy-Gourssat =Cdzazz022b 15 !1C 2C Hc9M_e wL 122200100 002,1sin2i sin ,CCzzCzdz zdzzz zz zzC+=vv = H = H Cauchys T01zC = H01222001|2izzCzdzzzzz= =v7201sin02iCzdzzz=v 02zC = H120102iCzdzzz=v702001sinsin | sin2izzCzdzzz=vb# b 16 !f ()zf 1|0 wL = Db T () ()zgzf = C k C
11、 = ) ( ) ( )zgzf = 9 b y () ()zgzf , D =)# C # =9) !0z C = B5Cauchys T ()( )()( )=CCdzzzzgizgdzzzzfizf000021,21 y C () ( )zgzf = # ( ) ( )=CCdzzzzgdzzzzf00 V7 () (),00zgzf = 0z i C = ( ( ) ( )zgzf = b 18 ! u D ()f z D =T_12KK 21KKc 0z 12KKW B k3.5.1 12().CKK+1 nm v 78: M ZEb 19 ! ()zf Y u D = O , C
12、D = BHe; wL s( )()Czfzf dz ,$ I $ ,by ()zf D =# ( )zf O f V7 ()zf D =9y D = () 0zf #( )()zfzf D = V7 C # C =9 Cauchy-Gourssat ()()=Cdzzfzf0b 20 k O C I V e wL$ 21 ! ()f z u D = C D = iBH_e wL D = C iB0z T200( ) ( )()CCfz fzdz dzzz zz=vv b Cauchy s T000( )2i()| 2i( )zzCfzdz f z f zzz=v 7 T0020() 2i(
13、) | 2 i ( )() 1!zzCfzdz fz fzzz=v# T b 22 T (, )x y (, )x y = O a Z 7yxs =xyt = +* ist+ ix y+ f b (, )x y (, )x y = 7yxs = xyt = + ,st B ,st C RZ Vbi 0xx yy + = 0xx yy + = 5 xyxx xyyyst =+= yyyxy x yx xst = =# ,st C R Z- 7 - ist+ ix y+ f b 23 ! u u D =f # iuufx y = f D =f $ I $ f D =f by u u D =f #x
14、u yu D =B b 2222xyuuuux xy y = 2yxuu ux xy y = C RZb 24f vxy=+ uxy=+ f $ I $ by iuv+ ?Bf b 25 ! u v f T v u f * u9 v f b $ I $ b I 275=b 26 B f f b ! v u f 5 0xx yyuu+ = 0xx yyvv+ = xyuv= yxuv= b () 2xx xx x x xxuv u v u v uv=+ + () 2yy yy y y yyuv u v u v uv= +# () () 0xx yyuv uv+=B f f b 27 T () i
15、f zuv=+ Bf k 1 i()f z 9 f 2 u v f 3222222 222| ( )| | ( )|4( ) 4 | ( ) |xxfz fzuv fzxy+=+=b 1 i() if zvu= 7 () if zuv=+ Bf # ,uv C RZ 7 ()xyvu= ()yxvu= b# i()f z 9 f b 2 () if zuv=+ Bf i() if zvu= b# u v f b 32222222222|()| |()|()()fz fzuvxyxy +=+ 222222 222222( )2( )4( ) 4 | ( ) |x x y y xx yy xx y
16、yxxuvuvuuu vvvuv fz=+ + +=+=28 22ux y=22yvx y=+ f iuv+ f b 2xux= 2yuy= 2222()xxyvx y=+22222()yx yvx y=+ 2223 22282()()xxxy yvx yxy=+3223 22286()()yyyyvx yxy=+5 2(2)0xx yyuu+=+=23223 223 2228880()()()xx yyxy y yvvxy xy xy+= + =+b - 8 - 29 p / T f u 1 (),u f ax by a b= + 2yufx=b 122, , ,xxx yuafu afu
17、bf= =7 0xx yyuu+ = 5 0f = 12()f cax by c=+b 22234 211, 2 , , ,xxx yyyyyufu ffufufxxxxx= = + = = 7 0xx yyuu+=5 2122120,arctanyy yf ffc cxx x+= + b 30/ Xf pf () if zuv= + 122()(4 )uxyx xyy= + + 222,(2) 0yvfxy= =+ 3 2( 1) , (2) iuxyf= = 4 arctan , 0yvxx= b 12222363, 363xyuxxyyuxxyy=+ =5 2222 2() i363i(
18、363)3(1)xyf z u u x xy y x xy y i z= = + = # 3() (1 ) i,fz iz cc= + 222 22 2222 222 222 2 222i1( ) i i()()()()yxx y xy x y xy zfz v vx yxyxyzzz =+= + = = =+ +# 111() , (2) 0 ()2fz c f fzzz= + = = 5 3 ( ) i 2 2i( 1) 2i( 1 i ) 2i( 1)xyfz u u y x x y z= = = += # 22() i( 1) , (2) i () i( 1)fz z c f fz z
19、= + = = 5 422 22 22i1( ) i iyxxyxyzfz v vx y xyxyzzz=+= + = = =+# () ln ,fz z cc= +b 31 ! sinpxve y= p p P vf i pf () if zuv= + b 2sin ( 1) 0pxxx yyvv e yp+= = 1p = b 1p = H () ,zfz e cc= + 1p = H() ,zfz e cc= + b 32 T (, )uxy u D =f C D =0z B_0|zz r = = c Db k 1 (, )uxy00(, )x y (, )uxy C ( 200 0 0
20、01(, ) ( cos, sin)2ux y ux r y r d =+ 2 (, )uxy00(, )x y (, )uxy00|zz r ( 0200 0 020001(, ) ( cos, sin)rux y ux r y r rd drr =+b 1 ( TP86 - 9 - 20001() ( e)2if zfzRd=+ | L 200 0 001(, ) ( cos, sin)2ux y ux r y r d =+ 2 10 020 0 00 002200 011(cos, sin) 2(,) (,)rrux r y r rddr uxyrdr uxy + = = b 33 T
21、() if zuv=+ u D =) C D =_ |z R= = c Db! z C =Bi 72/zRz= k 2() ()0CCfzfddzzR =vvb y z C =B22| /| /|RzRzR z RRz= #()fz C # =bCauchy 2() ()0CCfzfddzzR =vvb 34 Os T5 33T 22211 1( )()() ( )2i 2i ( )( )CCzRzff zfddzR z zR z =+ = vv C |z R= |. Os T 1()()2iCff zdz=v 7 33 5T2()0CzfdzR=v#| TMhb 35 T 7iie, eR
22、zr = 222/i.()( )()( )R2cos()dd dzR z z z Rr r = +i 345T 22 i201()(e)()22cosRrfRf zdRRr r=+. | L 22201( )(cos,sin)(, ) (cos , sin )22 )RruR Ruxy ur r dRRr r =+s Poissonsb YV TBf B = V V Ub 2iRRRRe=#22/.()( ) ()( )()( )dddRzR z z zzz = iidiRedidRe=22()( ) 2cos()zzRRr r = +# - 10 - 22/i()( ) 2cos()ddzz
23、RRr r = +b 3452i22 21( )() 1 ( )(e)()2i ( )( ) 2 2 cos( )CCRzzf RrfR dfz dzR z R Rr r = +vvb 36 ! ()f z e wL C =# C O n . 1 k Os T 1()()2innCff zdz=v. 2 ! M |()|f C Kv L C d z CK ks T3.1.10 1 T T 1/|()|2nLfz Md. 3 7 n + 2 T |K |()|f zM bTV u = f Kv ? uH | Kv b 1 Os T| f ()f z ()nf z b 2 11()|()|()|22nnn nCfLf zfz ds Mzd = v# 1/ 1/|()|22nnnLLfz M Mdd=b 3 2 T |Kn + b
