复变函数(第四版)课后习题答案

11i231+2i13ii1−−3()()2i5i24i3−+4i4ii218+−1()()()2i31312i32i32i32i31−=−+−=+133=⎭⎬⎫⎩⎨⎧+i231Re1322i31Im−=⎭⎬⎫⎩⎨⎧+,()2i31312i31+=[phone]i3122=⎟⎠⎞⎜⎝⎛−+⎟⎠⎞⎜⎝⎛=+kπ2i231argi231Arg+⎟⎠⎞⎜⎝⎛+=⎟⎠⎞⎜⎝⎛+,2,1,0,232arctan±±=+−=kkπ2()()()()i,25233i321ii)(1i1i13iiiii13ii1−=+−−−=+−+−−−=−−,23i13ii1Re=⎭⎬⎫⎩⎨⎧−−25i13ii1Im−=⎭⎬⎫⎩⎨⎧−−25i23i13ii1+=⎟⎠⎞⎜⎝⎛−−2342523i13ii122=⎟⎠⎞⎜⎝⎛−+⎟⎠⎞⎜⎝⎛=−−kπ2i1i3i1argi1i3i1Arg+⎟⎠⎞⎜⎝⎛−−=⎟⎠⎞⎜⎝⎛−−,±,±,=,+−=210235arctankkπ.3()()()()()()()()()42i7i262i2i2i5i24i32i5i24i3−−=−−−+=−+13i27226i7−−=−−=()()272i5i24i3Re−=⎭⎬⎫⎩⎨⎧−+()()132i5i24i3Im−=⎭⎬⎫⎩⎨⎧−+1()()l3i272i5i24i3+−=⎥⎦⎤⎢⎣⎡−+()()22952i5i24i3=−+()()()()kππkπ2726arctan22i2i52i43argi2i52i43Arg+−=+⎥⎦⎤⎢⎣⎡−+=⎥⎦⎤⎢⎣⎡−+(),2,1,0,12726arctan±±=−+=kkπ.4()()()()ii141iii4ii4ii10410242218+−−−=+−=+−3i1i4i1−=+−={}{}3i4iiIm1,i4iiRe218218−=+−=+−3i1i4ii218+=⎟⎠⎞⎜⎝⎛+−10|i4ii|218=+−()()()2kπ3i1arg2kπi4iiargi4iiArg218218+−=++−=+−=.2,1,0,k2kπarctan3±±=+−2()i13i53yi1x+=+−++x,y()()[]()()()3i53i53i53yi1x3i53yi1x−+−−++=+−++()()()()[]343y51x3i3y31x5−++−+−++=[]()i1185y3xi43y5x341+=−+−+−+=⎩⎨⎧=−+−=−[phone]yxyx⎩⎨⎧=+−=[phone]yxyx11,1==yx3i-i=i-1=i421)||116)Re()(),Im()()22izzzzzzzz=z=+=#−2izxy=+5z2||zz=222zizxy=+2||zz=2222ixyxyx−+=+y02222,xyxyxy−=+=z6na1||≤z||azn+|a||a||z|aznn+≤+≤+1naezargi=()|a|ea|a|eea|zaannan+=+=+⎟⎟⎠⎞⎜⎜⎝⎛=+|11argiargiargi||1a+81i2-131+3i4()π0isincos1≤≤+−ϕϕϕ5i12i+−6()()32isin3cos3isin5cos5ϕϕϕϕ−+12πie2πisin2πcosi=+=2iπeisinπcosπ1=+=−33πi2e3πisin3πcos223i2123i1=⎟⎠⎞⎜⎝⎛+=⎟⎟⎠⎞⎜⎜⎝⎛+=+421cosisin2sini2sincos2sinsinicos222222ϕϕϕϕϕϕϕϕ⎛⎞−+=+=+⎜⎟⎝⎠π)(0,e22sin2πisin2πcos22sin2πi≤≤=⎟⎠⎞⎜⎝⎛−+−=−ϕϕϕϕϕϕ5()⎟⎟⎠⎞⎜⎜⎝⎛−=−=−−=+−21i212i1i12i21i12i⎟⎠⎞⎜⎝⎛−=4πisin4πcos2=4πie2−6()()()()223i5i3i10i9i193cos5isin5e/ee/eecos3isin3ϕϕϕϕϕϕϕϕ−−+==−ϕ=3ϕϕisin19cos19+=911111,;xxayyb=+⎧⎨=+⎩21111cossin,sincos.xxyyxyαααα=−⎧⎨=+⎩11iAab=+11izxy1=+izxy=+121zzA=+i11(cosisin)ezzzααα=+=10-izezzArgi||=()⎟⎠⎞⎜⎝⎛−−=⋅||=−2Argi2iArgiπzπz|z|eeeziz2π11222121212||||2(|||zzzzzz++−=+2|)2212121212121211222212||||()()()(2()2(||||)zzzzzzzzzzzzzzzzzz++−=+++−−=+=+)121()()()PzRzQz=iXY+XYxy2()Rz1()iRzXY=−3iab+10110nnnnazazaza−−++++=iab−1()()()Re(()())Im(()())()()(,)(,)()()PzPzQzPzQzPzQzRzQzqxyqxyQzQz===+;2()()()()ii()()()PzPzPzRzXQzQzQz⎛⎞YXY====+=−⎜⎟⎝⎠;3()1011nnnnPzazazaza−−=++++4()zPzazazaann=++++=221013itez=1ntzznncos21=+2ntzznnsini21=−1nteeeezznnsin21intintintint=+=+=+−2nteeeezznnsini21intintintint=−=−=−−141(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⎟⎠⎞⎜⎝⎛+=45sini45cos2264/5i6πππe15(1i)(1i)nn+=−n5i/4i/4i/4i/4(2e)(2e),eennnnππππ−−==sin04nπ=4,0,1,2,nkk==±±161083=+z208'''=+yy1()()1i123382kzeπ+=−=k=0,1,2,3i1+,2−3i1−2083=+λi311+=λ22−=λi313−=λ()xCxCeeCyxx3sin3cos3221++=−17z111,,,,,zzzzzz−−−oxyz-zzz−1z1z1z−181z2z321,,zzz1()2121zzz+=2()211zzzλλ−+=λ3()32131zzzz++=1,2,3i=+=,kyxzkkk12i22121yyxxz+++=z1z2z62()()[]122122iyyλyxxλxz−−+−−=1z2z|z|z|z|z121λ−−=3()(3213213i31yyyxxxz+++++=)z321zzz∆19123,,zzz0321=++zzz,1321===zzzz1z2z31=z1321===zzz321zzz∆2331zz==3z()[]()[]212322112121zzzzzzzzzzzz+++=+−+−=21212zzzz++=12121−=+zzzz)())((122122112121221zzzzzzzzzzzzzz+−+=−−=−()322121=+−=zzzz321=−zz33231=−=−zzzz321zzz∆1=z20z1z2z332311312zzzzzzzz−−=−−321312zzzzzz−=−=−)arg()arg()arg()arg(32311312zzzzzzzz−−−=−−−231312zzzzzz∠=∠()()()()1232321331121312zzzzzzzzzzzzzzzz−−=−+−−+−=−−123312zzzzzz∠=∠321zzz∆321312zzzzzz−=−=−721z1|52|z−=61|i2|≥+z3Re(2)1z+=−4()3iRe=z5|i||i|−=+zz64|1||3|=+++zz7Im()2z≤8123≥−−zz90argzπ10()4iargπ=−z105z=6a2i20−=z1b3Re(2)13zx+=−⇔=−z3x=−c4xyyxzi)i(ii+=−=.33)iRe(=⇔=yzzy=3d522iiiii)(i)(i)(i)zzzzzzzz+=−⇔+=−⇔+−=−+⇔22ii1ii1ii02Re(i)020zzzzzzzzzy−++=+−+⇔−=⇔=⇔=0.y⇒=ze6222214)2(122)14(3413+=−⇔+−=−⇔+−=+⇔=+++zxzxzzzz134)2041232222=++⇔=++⇔yxyxz-3,0-1,023f7g2y≤8≥+−−⇔−−≥−−⇔−≥−⇔≥−−933)2)(2()3)(3(23123222zzzzzzzzzzz252245.z2zzzzzx−−+⇔+≤⇔≤25=x25=xh9i10i0,1+=xxyj8xy-2(b)Oix5(a)yO-3(d)y3ixO(c)xyz-iiyxi3-2Oy5/2xxy=x+1iyO(e)(f)(g)(h)(i)2i(j)22120Imz41−z31Re0z423z≤≤531+−zz61arg1zπ−−+97141+−zz8|2||2|zz6−++≤92|2|1zz−−+10(2i)(2i)4zzzz−+−−≤10ImzyOx241−zxy5O116)1(22=+−yz301Rezx=0x=1xOy23Oxy423z≤≤10235131−⇔+−xzzy-1DOxx=-161arg1zπ−−+xyo-11=θπθ+=172221781411515zzxy⎛⎞⎛−+⇔++⎜⎟⎜⎝⎠⎝⎞⎟⎠OxD8/15-17/15y111517−=z1588|2||2|zz−++≤6xyo3522195xy+=92242|2|141,015zzxyx−−+⇔−D1/2yx224415xy1−=10(22i)(i)4zzzz−+−−≤xyo2-112923z22(2)1xy−++=Czaza=+aCCByAx=+)(i21Im),(21Rezzzyzzzx−==+==CzBAzBA=−+−)i(21)i(21)i(21BAa=+(21a=)iBA−Czaza=+240,(zzzzc)cααα+++=22()2()0zazaRzzazazaaR++=⇔+++−=caaR=−25t12tbtazsinicos+=tzi)1(+=3ttzi+=422ittz+=567zeabα==+1tαchishzatbt=+iittzaebe−=+,(i)⎩⎨⎧∞−∞==⇔+=+=ttytxtyxz,)i1(ixy=2sinsinicosi⎩⎨=⇔+=+=bytbtayxzπ20,cos≤⎧=tttax122b22=+yax3⎪⎩⎪⎨=⇔y11⎧=+=+=ttxttyxzii=xy⎪⎩⎪⎨⎧==⇔+=+=22221iitytxttyxz41=xy132222chchish1shxatxyzatbtybtab=⎧=+⇔⇔−=⎨=⎩522221,()()xyabab+=+−62arctan2