复变函数习题解答-3

-1-1.∫+idzz3021i3+233i3+3iii3+1⎩⎨⎧==,,3tytx10≤≤tttzi3+=10≤≤t()dtdzi3+=()()dtttdzzi3i32130102++=∫∫+()∫+=1023i3dtt()i3266i33101|i)3(31333+=+=+=t2∫∫∫∫++++=i30i30222221dzzdzzdzzdzzCC1C⎩⎨⎧==,,3tytx()10≤≤t2C⎩⎨⎧==,,3tyx()10≤≤t()∫∫∫++=⋅++⋅=i301010222i3266ii339dttdttdzz3∫∫∫∫∫+++=+=i30i0i3i2222243dzzdzzdzzdtzdzzCC()10i:3≤≤=ttzC()10i3:4≤≤+=ttzC()∫∫∫++=⋅++⋅−=i301010222i32663i3idttdttdzz2xy=2xy=()∫++idzyx102i1xy=()10i≤≤+=tttz()dtdzi1+=()()()∫∫+++=+i101022i1iidtttdzyx()()()∫+−=⎟⎠⎞⎜⎝⎛++=++=102i65612i31i1ii1dttt22xy=()10i2≤≤+=tttz()dttdz2i1+=()()()∫∫+++=+i10102222i1iidttttdzyx()()()()∫∫++=++=10103222ii12i1i1dtttdttt()i65612i31i1+−=⎟⎠⎞⎜⎝⎛++=3()zfDCD()[]()[]∫∫==0ImRedzzfdzzfCC()zzf=1:=zC()zfx3+iC2C1OC3iC4y(z)3-2-()[][]∫∫=πθθ20ReReiiCdeedzzf()∫≠=+−=ππθθθθ200icosisincosd()[][]∫∫=πθθ20iiImImdeedzzfC()∫≠−=+−=ππθθθθ200cosisinsind41zz=2iCzdzπ=∫vC||1z=12iCCzdzdzzπ==∫∫vv5dzzzC∫C12=z24=z12||=z2||=z4||2==⋅zzzzz4=i422||2||2||4π===∫∫∫==dzzdzdzzzzCzZ2C4||=z16||2==⋅zzzzz16=i844||4||4||16π===∫∫∫==dzzdzdzzzzCzZ671∫−Czdzze21|2:|=−zC222Cdzza−∫v:||Czaa−=3i21zCedzz+∫v:|2i|3/2Cz−=43Czdzz−∫v:||2Cz=523,(1)(1)Cdzzz−−∫v:||1Czr=63cosCzzdz∫v0Cz722(1)(4)Cdzzz++∫v:||3/2Cz=8sinCzdzz∫v1|:|=zC9∫⎟⎠⎞⎜⎝⎛−Cdzzz22sinπ2|:|=zC105zCedzz∫v1|:|=zC1Cauchy∫==−=Czzzeedzzei2i2222ππ21∫∫=+=−+=−=CCazaazdzazazazdzi1i2122ππ2∫∫∫⎥⎦⎤⎢⎣⎡+−−=−CCCdzazdzazaazdz112122[]i0i221aaππ=−=3Cauchyiii2i/(i)2i/1-iizzzCCzedzedzzeezzzππ=+===++∫∫vv-3-45607i±=zCi2±=zCCauchy∫∫∫=+=−+++++=++31|i|2231|i|2222)4)(1()4)(1()4)(1(zCzzzdzzzdzzzdz=()()()()∫∫=+=−++++−++31|i|231|i|2i4i1i4i1zzdzzzzdzzzzi2i2)4i)((1i2)4i)((1i2−==+−+++=zzzzzzππ033=−=ππ8Cauchy0sin2isin|0zCzdzzzπ===∫v9()0'sini22sin22==⎟⎠⎞⎜⎝⎛−=∫πππzCzdzzz10(4)052ii()|4!12zzzCedzezππ===∫v813i2izedzππ−∫20i6ch3zdzπ∫3i2-isinzdzππ∫410sinzzdz∫5i0(i)zzedz−−∫6i211tan(1i)coszdzz+∫13i23i2ii02zzeedzππππ−−=∫200i/6i61ch3sh3|i/33zdzzππ==−∫3ii2i-i-i-i1cos2sin21sin()|(sh2)i2242zzzzdzdzππππππππ−==−=−∫∫41100sin(sincos)|sin1cos1zzdzzzz=−=−∫5ii00(i)(i1)|1cos1i(sin11)zzzedzze−−−=−−=−+−∫6i2i221211tan11(tantan/2)|(tan1tan1th1)ith1cos22zdzzzz+=+=−+++∫9143(),:||412iCdzCzzz+=++∫v222i,:|1|61CdzCzz=+∫v312123cos,:||2:||3CCCzdzCzCzz=+==∫v-4-416,i25CdzCzi±±∫v53,||1:||1()zCedzaaCzza≠=−∫v143()2i(43)14i12iCdzzzππ++=++∫v22|i|1|i|12i2/(i)2/(i)01-iiCzzizizdzdzdzzzz−=+=+−=+=++∫∫∫vvv3121200333coscoscos2i2i(cos)''|(cos)''|02!2!zzCCCCCzzzdzdzdzzzzzzππ−===+=−=−=∫∫∫vvv42iiCdzzπ=∫v5||1a331/()||10()zCezazdzza−≤−∫v||1a32i()''|i()2!zzazaCedzeezaππ==−∫v10C210Cdzz=∫vC0212i(1)'|0zCdzzπ===∫vC21/zC210Cdzz=∫v11211||2zzdzz=∫v2||4zzdzz=∫v2i||202i0zzdzedzπθθ−===∫∫v2i||404i0zzdzedzπθθ−===∫∫v1zz12DzD||1z=DCz201Re.14zdπζζ⎡⎤=⎢⎥+⎣⎦∫211ζ+0zCi1222i000011i2icos.111422cos2zeddxddxeηθθηπηζηηζη=+=+++++∫∫∫∫2i1eη−+-5-B1C1C2B2MNEFBGH201Re.14zdπζζ⎡⎤=⎢⎥+⎣⎦∫131C2CMN1B2B1B2BB()fz1BB−2BB−1C2C12()()CCfzdzfzdz=∫∫vv1BB−()fz()0MENGMfzdz=∫v()0MHNFMfzdz=∫v()()()()NGMMENMHNNFMfzdzfzdzfzdzfzdz+=+∫∫∫∫12()()CCfzdzfzdz=∫∫vv14Ca-aaa-aC∫−Cdzazz22iaC-aC∫∫=+=−+=−=CazCazzdzazazzdzazzii222ππiia−CaC∫∫=−=+−=−−=ccazazzdzazazzdzazzii222ππiiiaaC1C,2Caa−,21,CCCCauchy∫∫∫=+=+−+−+=−ccciiidzazazzdzazazzdzazz21222πππivaaCCauchy-Gourssat∫=−Cdzazz022151C2C122200100002,1sin2isin,CCzzCzdzzdzzzzzzzCπ⎡⎤⎧+=⎢⎥⎨−−⎢⎥⎩⎣⎦∫∫vvCauchy01zC01222001|2izzCzdzzzzzπ===−∫v201sin02iCzdzzzπ=−∫v02zC120102iCzdzzzπ=−∫v02001sinsin|sin2izzCzdzzzzzπ===−∫v16()zf1||0zCrz=||10r()zfz=0()21zzf=1||0zCrz=||10r-6-()∫∫===rzCdzzdzzf||201()21zzf=z=017.()zf()zgDCDD()()zgzf=CC()()zgzf=()()zgzf,DC0zCCauchy()()()()∫∫−=−=CCdzzzzgizgdzzzzfizf000021,21ππC()()zgzf=()()∫∫−=−CCdzzzzgdzzzzf00()(),00zgzf=0zC()()zgzf=18D()fzD12KK21KK0z12KK3.5.112().CKK−+7819()zfDCD()()∫Czfzf'dz()zfD()zf()zf'DD()0≠zf()()zfzf'DCCCauchy-Gourssat()()∫=Cdzzfzf0'20C21()fzDCDDC0z200'()()()CCfzfzdzdzzzzz=−−∫∫vvCauchy000'()2i'()|2i'()zzCfzdzfzfzzzππ===−∫v0020()2i'()|2i'()()1!zzCfzdzfzfzzzππ===−∫v22(,)xyϕ(,)xyψyxsϕψ=−xytϕψ=+ist+ixy+(,)xyϕ(,)xyψyxsϕψ=−xytϕψ=+,st,stCR0xxyyϕϕ+=0xxyyψψ+=xyxxxxyyyystϕψϕψ=−=+=yyyxyxxyxxstϕψϕψ=−=−−=−,stCR-7-ist+ixy+23uDiuufxy∂∂=−∂∂fDfDuDxuyu−D2222xyuuuuxxyy⎛⎞∂∂∂∂⎛⎞==−=−⎜⎟⎜⎟∂∂∂∂⎝⎠⎝⎠2yxuuuxxyy⎛⎞∂∂∂⎛⎞==−−⎜⎟⎜⎟∂∂∂∂⎝⎠⎝⎠CR24vxy=+uxy=+iuv+25uvvuuv2726vu0xxyyuu+=0xxyyvv+=xyuv=yxuv=−()2xxxxxxxxuvuvuvuv=++()2yyyyyyyyuvuvuvuv=++()()0xxyyuvuv+=27()ifzuv=+1i()fz2u−v3222222222|()||()|4()4|'()|xxfzfzuvfzxy∂∂+=+=∂∂1i()ifzvu=−()ifzuv=+,uvCR()xyvu=−()yxvu=−−i()fz2()ifzuv=+i()ifzvu=−u−v3222222222222|()||()|()()fzfzuvxyxy∂∂∂∂+=++∂∂∂∂222222222222()2()4()4|'()|xxyyxxyyxxyyxxuvuvuuuvvvuvfz=+++++++=+=2822uxy=−22yvxy=+iuv+2xux=2yuy=−2222()xxyvxy−=+22222()yxyvxy−=[phone]()()xxxyyvxyxy=−++322322286()()yyyyvxyxy=−++2(2)0xxyyuu+=+−=232232232228880()()()xxyyxyyyvvxyxyxy+=+−=+++-8-29u1(),ufaxbyab=+2yufx⎛⎞=⎜⎟⎝⎠122','','',xxxyyuafuafubf===0xxyyuu+=''0f=12()fcaxbyc=++22234211',2''',','',xxxyyyyyyufuffufufxxxxx=−=+==0xxyyuu+=21221''2'0,arctanyyyfffccxxx⎛⎞++==+⎜⎟⎝⎠30()ifzuv=+122()(4)uxyxxyy=−++222,(2)0yvfxy==+32(1),(2)iuxyf=−=−4arctan,0yvxx=12222363,363xyuxxyyuxxyy=+−=−−22222'()i363i(363)3(1)xyfzuuxxyyxxyyiz=−=+−−−−=−3()(1)i,fzizcc=−+∈\2222222222222222222i1'()ii()()()()yxxyxyxyxyzfzvvxyxyxyzzz−−−−=+=+===+++111(),(2)0()2fzcff