复变函数与积分变换【参考答案】-王忠仁-张静著-高等教育出版社-62

1.1z2||zz=222zizxy=+2||zz=2222ixyxyx−+=+y02222,xyxyxy−=+=z11.21(5i3−)2()6i1+361−4()31i1−1()()6/5i56/i553222i232i3ππ−−==⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛−=−ee5π5π32cosisin16316i66⎡⎤⎛⎞⎛⎞=−+−=−−⎜⎟⎜⎟⎢⎥⎝⎠⎝⎠⎣⎦2()()666i/43i/21i1i22e8e8i22ππ⎡⎤⎛⎞+=+===−⎢⎥⎜⎟⎝⎠⎣⎦3()()1iπ21/6iπ+2661ee,0,1,2,3,4,5kkkπ+−===61−6,2i23e/6i+=πie/2i=π2i23ei/65i+−=π2i23e/6i7−−=πi23i−=/πe2i23411i−=/πe4()()0,1,2=,==⎥⎦⎤⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛2−212=−13⎟⎠⎞⎜⎝⎛+−31/−3131keekπππ24i64i22ii3()1/31i−,127sini127cos22,12sini12cos22612/7i662/i6⎟⎠⎞⎜⎝⎛+=⎟⎠⎞⎜⎝⎛−=−ππππππee⎟⎠⎞⎜⎝⎛+=45sini45cos2264/5i6πππe21.31083=+z1()()1i123382kzeπ+=−=k=0,1,2,3i1+,2−3i1−1.4z1|52|z−=61|i2|≥+z3Im()2z≤40argzπ105z=6a2i20−=z1b3c2y≤4dxy-2(b)Oix5(a)yO(c)2i(d)1.5120Imz41−z31Re0z423z≤≤10ImzyOx241−zxy5O116)1(22=+−yz301Rezx=0x=1xOy23Oxy423z≤≤2331.63zw=1i1=zi12+=zi33+=z2w3arg0πzwθirez=θω333ierz==422i14sini4cos2i1,iπππzez=⎟⎠⎞⎜⎝⎛+=+===i2πe16i3223z⎜⎜⎝=26sini6cos221i3iπππe=⎟⎠⎞⎜⎝⎛+=⎟⎟⎠⎞⎛+=+4wi3/3i1−==πew2i221323⎟⎞ٛi2122242+−=⎟⎠⎜⎜⎝⎛+−==πewi822i33==πew2πwarg01.71(),(0)2izzfzzzz⎛⎞=−≠⎜⎟⎝⎠0z→()fz2x212()2izzxyfzzzy⎛⎞=−=⎜⎟+⎝⎠1.8)arg(argππ≤−zzzzzfarg)(=f(0)f(z)z=000z0)00(=xxz⎩⎨⎧−=⎪⎪⎟⎠⎜⎝−→πxxxarctanlim0⎪⎩⎪⎪⎪⎨⎧⎞⎛⎟⎠⎞⎜⎛+=−→→→→πππyyzyyxzzarctanlimarglim00arg0⎝+xx00zargargzzzlim→zzf(z01)z0(zf0)≠z00fzfzz0)(≠zf)(0z)(lim=→0)(0≠zf02)(0=zfε0∃δδ−0zz2)()()(00zfzfzf=−ε)(20)(0zf0)(zfzf−20)(0zf)(zf),(δzUz∈20)(≠zf0lim()zzfzA=→()fz0z1ε=0δ00||zzδ−|()|1fzA−≤00||zzδ−|()||()||()|||1||fzfzAAfzAAA=−+≤−+≤+562.112111)()',()2'nnznznzz−⎛⎞==−⎜⎟⎝⎠11221100()()'limlim()nnnnnnnnzzzzzznzCzzznzz−−−−∆→∆→+∆−==+∆+∆=∆220011111'limlim()zzzzzzzzzzz∆→∆→−⎛⎞+∆==−=−⎜⎟∆+∆⎝⎠2.21()yxzfi2−=233()23ifzxy=+3()yxxyzf22i+=4()sinchicosshfzxyxy=+11,0,0,2−=∂∂=∂∂=∂∂=∂∂yvxvyuxxuz21−=xu,vC-R()yxvuzfii−=+=21−=xz226uxx∂=∂0uy∂=∂0vx∂=∂29vyy∂=∂z2223,230xyxy=±=u,vC-R()33i23ifzuvxy=+=+230xy±=z32yxu=∂∂xyyu2=∂∂xyxv2=∂∂2xyv=∂∂zz=0u,vC-R()yxxyzf22i+=0=zz4coschuxyx∂=∂sinshuxyy∂=∂sinshvxyx∂=−∂coschvxyy∂=∂zu,vC-R()sinchicosshfzxyxy=+zz2.3()fz15(1)z−232izz+3211z−4(,0)azbcdczd++1()45(1)fzz′=−()zfz2()i232+=′zzf()zfz3()()()()222211212+−−=−−=′zzzzzzf()zf1±=zz1±=z()zf74()2()adbcfzczd−′=+()zf/(0)zdcc=−≠110z0zDD2221()zf0z()0zf′2()0zf′()zf0z30z()zf()zf0z40z()zf()gz0z()()fzgz+()/()fzgz5(,)uxy(,)vxy()ifzuv=+6()ifzuv=+u()fzDv()fzD1()222||yxzzf+==zz=02()2||zzf=z=0z=0300()()fzzzfz4()sinch,()icosshfzxygzxy==(/2,0)zπ=()()fzgz+5()xyxzzzfiRe2+==z=0C-R()zfz=06uC-Rv()fzD2.5()vuzfi+=z()()()222|'|||||zfzfyzfx=⎟⎟⎠⎞⎜⎜⎝⎛∂∂+⎟⎠⎞⎜⎝⎛∂∂()22||vuzf+=()fzD()fz0z()fzD()fz0z()fz0z2.48()22||vuxvvxuuzfx+∂∂+∂∂=∂∂()22||vuyvvyuuzfy+∂∂+∂∂=∂∂()vuzfi+=yvxu∂∂=∂∂xvyu∂∂−=∂∂()()⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛∂∂−+⎟⎠⎞⎜⎝⎛∂∂+=⎟⎟⎠⎞⎜⎜⎝⎛∂∂+⎟⎠⎞⎜⎝⎛∂∂222222221||||xvuxuuvuzfyzfx⎥⎥⎦⎤∂∂⎟⎠⎞⎜⎝⎛∂∂−+∂∂∂∂+⎟⎠⎞⎜⎝⎛∂∂+⎟⎠⎞⎜⎝⎛∂∂+xuxvuvxvxuuvxvvxuv222222()()()22222222222222||||11zfzfvuvuxvxuvxvxuuvu=++=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛∂∂+⎟⎠⎞⎜⎝⎛∂∂+⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛∂∂+⎟⎠⎞⎜⎝⎛∂∂+=2.7-θ∂∂=∂∂vrru1θ∂∂−=∂∂urrv1θθsin,cosryrx==vu,C-Rθθsincosyuxuru∂∂+∂∂=∂∂()rurrxuryuryvrxvv∂∂=∂∂+∂∂=∂∂+−∂∂=∂∂θθθθθcossincossinθ∂∂=∂∂vrru1()θθθcossinryurxuu∂∂+−∂∂=∂∂θθθθsincossincosxuyuyvxvrv∂∂+∂∂−=∂∂+∂∂=∂∂θθθ∂∂−=⎟⎟⎠⎞⎜⎜⎝⎛∂∂−∂∂−=urrxuryur1sincos1θ∂∂=∂∂vrru1θ∂∂−=∂∂urrv12.8()ivuzf+=D()zf1()zf2()zfD3()||zfD4()zfargD5cbvau=+abc91()zf0=v()zfDC-R0=∂∂=∂∂yvxu0=∂∂−=∂∂xvyuuD()Cu=()Civuzf=+=2()ivuivuzf−=+=D()yvyvxu∂∂−=∂−∂=∂∂()xuxvyu∂∂=∂−∂−=∂∂1()ivuzf+=Dyvxu∂∂=∂∂xvyu∂∂−=∂∂2120=∂=∂∂=∂∂=∂∂vyvxvyuxuvu,D()212211,,CCCuCu==()CiCCivuzf=+=+=213()||zfD1C2122Cvu=+xy⎪⎪⎩⎪⎪⎨⎧=∂∂+∂∂=∂∂+∂∂022022yvvyuuxvvxuu()zfDC-Rxvyuyvxu∂∂−=∂∂∂∂=∂∂,⎪⎪⎩⎪⎪⎨⎧=∂∂+∂∂=∂∂−∂∂00yuuxuvyuvxuu0=∂∂=∂∂yvxu0=∂=∂∂vyvxvvu,21,CvCu==()CiCCivuzf=+=+=214zargD1C()0≠zf()0≠+=ivuzf()⎪⎪⎪⎩⎪⎪⎪⎨⎧−+=0,0,arctan0,0,arctan0,arctanargvuuvvuuvuuvzfππ⎪⎩⎪⎨⎧−+=0,00,01111vuCvuCuCππ()zfargxy100112222=+∂∂−∂∂=⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛∂∂−∂∂vuxuvxvuuvxuvxvuu0112222=+∂∂−∂∂=⎟⎠⎞⎜⎝⎛+⎟⎟⎠⎞⎜⎜⎝⎛∂∂−∂∂vuyuvyvuuvyuvyvuu()zfC-R⎪⎪⎩⎪⎪⎨⎧=∂∂−∂∂=∂∂−∂∂−00yuvxuuyuuxuv0=∂∂=∂∂yuxu0=∂∂=∂∂yvxuuv2C3C()CiCCivuzf=+=+=325cbvau=+abcab0022≠+baabccbvau=+xy⎪⎪⎩⎪⎪⎨⎧=∂∂+∂∂=∂∂+∂∂00yvbyuaxvbxua()ivuzf+=uvC-R⎪⎪⎩⎪⎪⎨⎧=∂∂+∂∂=∂∂−∂∂00yuaxubyubxua0=∂∂=∂∂yuxu0=∂∂=∂∂yvxv()zf2.9301=+ze431−=ze()()[]()kkz21i21argi|1|ln1Ln+=+−+−=−=ππ,2,1,0±±=k12shiz=4shiz=ize=1Lnii(2),0,1,2,2zkkπ==+=±±sinz=0cosz=0(1)设sinz=0,则eiz−e−iz2i=0,从而，(eiz)2=1.即：e2iz=1.注意：1=e0+2kπi故z=kπ112.10()iLn−()i43Ln+−()()()⎟⎠⎞⎜⎝⎛+−=+−+−=−πππkk22i2iargi|i|LniLn,2,1,0,212i±±=⎟⎠⎞⎜⎝⎛−=kkπ()()2iiargi|i|lnilnπ−=−+−=−()()[]πk2i43argi|i43|lni43Ln++−++−=+−⎥⎦⎤⎢⎣⎡+⎟⎠⎞⎜⎝⎛−+=ππk234arctani5ln2.1111212Ln()LnLnzzzz=+21212Ln(/)LnLnzzzz=−11212121212Ln()ln(||)iArglnlniArg+iArgzzzzzzzzzz=+=++12LnLnzz=[phone]Ln(/)ln(|/|)iArg/lnlniArg-iArgzzzzzzzzzz=+=−+12LnLnzz=−2.122i1π−e1iexp4π⎛⎞⎜⎟⎝⎠i3()ii1+eeeeei2sini2cos2i2i1−=⎟⎠⎞⎜⎝⎛−==−−ππππ()111i44441i2expcosisin1i4442eeeeππππ⎛⎞⎛⎞===⎜⎟⎜⎟⎝⎠⎝⎠()iln3iarg32iiLn32iln33kkeeeeππ++⎡⎤−⎣⎦===(),2,1,0,3lnsini3lncos2±±=+=−kekπ()()[]()()iiln|1i|iarg1i2iLn1i1ikeeπ++++++==⎟⎠⎞⎜⎝⎛+==⎟⎠⎞⎜⎝⎛+−⎟⎠⎞⎜⎝⎛+−22lnsini22lncos2412422lnikkeeπππ,2,1,0±±=k-12-3.1.∫+idzz3021i3+233i3+1⎩⎨⎧==,,3tytx10≤≤tttzi3+=10≤≤t()dtdzi3+=()()dtttdzzi3i32130102++=∫∫+()∫+=1023i3dtt()i3266i33101|i)3(31333+=+=+=t2∫∫∫∫++++=i30i30222221dzzdzzdzzdzzCC1C⎩⎨⎧==,,3tytx()10≤≤t2C⎩⎨⎧==,,3tyx()10≤≤t()∫∫∫++=⋅++⋅=i301010222i3266