复变函数与积分变换北京邮电大学课后习题答案.pdf

? 1 / 37 ? 1. ? a+ib ? /4 3513 ;(2)(43 ); 711 i i eii iii + + + . ? i 4 2222 ecosisinii 442222 =+=+ = ? ()() ()() 35i 17i35i1613 i 7i11+7i 17i2525 + = + + ? ()()2i43i834i6i5 10i+=+=+ ? ()3 1i1335 =ii i1i222 + += + 2.?(z=x+iy) ( za a za + ); 33 31313 ; . 22 nii zi + ? ? z=x+iy ? () () () () ()() () 2 2 iiii ii xayxayxyaxay za zaxyaxay xay + = + + ? () 222 2 2 Re zaxay za xay = + + , () 2 2 2 Im zaxy za xay = + + ? ? ? z=x+iy ? ()() ()()() ()() () 32 322 222222 3223 iii2ii 22i 33i zxyxyxyxyxyxy x xyxyy xyx y xxyx yy =+=+=+ =+ =+ ?() 332 Re3zxxy=, () 323 Im3zx yy=? ? ? () ()()()() 3 3 23 2 1i 3 1i 31 13133133 288 + + = + () 1 80i1 8 =+= ? 1i 3 Re1 2 + = , 1i 3 Im0 2 + = ? ? ? ()()()()() 23 32 3 13133133i 1i 3 28 + + = () 1 80i1 8 =+= ? 1i 3 Re1 2 + = , 1i 3 Im0 2 + = ? ? ? () () 1, 2 i 21 1i, k n k nk k nk = = =+ ? ?2nk=? ( )() Re i1 k n = ? ( ) Im i0 n = ? ?21nk=+?( )Re i0 n =? ( )() Im i1 k n = ? 3.? 1 2;3;(2)(32 );. 2 i iii + + ?2i415 +=+=? 2i2i + = ?33= 33 = ?( )()2i32i2i 32i51365+=+= ? ()()() ()() ()2i32i2i32i2i32i47i+=+= ? 1i 1i2 222 + + = ()1i11i 222 i + + = 4?zz=?z? ?zz=?izxy=+? ? 2 / 37 ? iixyxy+=?()2i0y=?y=0 ?z=x? ?z=x?x?zxx=? ?zz=? ? 5?z,w? zwzw+? ?() ()()( ) 2 zwzwzwzwzw+=+=+ () () 22 22 2Re z zz ww zw w zzwz ww zwz w =+ =+ =+ () 22 22 2 2 2 zwzw zwzw zw + =+ =+ ? ?zwzw+? 6?z,w? () 222 2Rezwzz ww+=+ () 222 2Rezwzz ww=+ () 2222 2zwzwzw+=+ ? ? () 222 2Rezwzz ww+=+? ? ? () 222 2Rezwzz ww=+? ?() ()()( ) 2 22 zwzwzwzwzw zz ww zw = =+ () 22 2Rezz ww=+? ? () 2222 2zwzwzw+=+ ? ? 7.? 3 3522 ;1;8(13 );.cossin 7199 i iii i + + + ? ()() ()() 35i 17i35i 7i117i 17i + = + 3816i198i17 e 50255 i =? 8 arctan 19 =? ?eii =? 2 =? 2 e i i = ? ii 1ee = ? () 2 8 13i16 3 += . ? () 2i 3 8 13i16 e += ? 3 22 cosisin 99 + ? 3 22 cosisin1 99 += ? ? 3 22 i.3i 93 22 cosisin1 ee 99 += = 8.?(1)i?(2)-1?(3) 33i+ ?. ?i? ? () 1 3 3 2 2 22 icossincosisin0,1,2 2233 + =+=+= kk ik ? 1 31 cosisini 6622 =+=+z? 2 5531 cosisini 6622 =+= +z 3 9931 cosisini 6622 =+= z ?-1? ? ()() 1 3 3 2 +2 1cosisincosisin0,1,2 33 kk k + =+=+= ? 1 13 cosisini 3322 =+=+z 2 cosisin 1=+= z ? 3 / 37 3 5513 cosisini 3322 =+= z ?33i+? ? i 4 22 33i= 6i6 e 22 += ? () () 1 1 2 i 44 2 2 44 33i6 e6cosisin0,1 22 kk k + +=+= ? 11 i 844 1 6cosisin6e 88 =+= z 911 i 844 2 99 6cosisin6e 88 =+= z? 9.? 2 e,2 i n zn=. ? 1 10 n zz +=L ? 2 i e n z = ?1 n z =?10 n z =? ?( )() 1 1 10 n zzz +=L ?n?2? ?z?1 ? 21 1+0 n zzz +=L 11.? :,0,e . i zrracr zc =+ ? :Im0 za Lz b = , ?eib =.?L?a? ?. ? ?L=z: Im za b =0?a? ?b?a? ?CA?L?C?L?BCD=? ?ACB=90 ?-=90 ?L?T? ?-=90? 12.?z? ?. (1)arg; (2); 1 (3)1| 2; (4)ReIm ; (5)Im12. z zz zi zz z z = = 1?|z| ?. ? n称以k ?, ? 1 (1) ln 2 k k k = ? ? n称以k+令 ?, ? 1 ( 1) ln(21) k k k = + ? ? (5) 0000 cosi1ee1e11 ( )() 2222222 nn nn nn nnnn n e = + =+ ? 0 e ( ) 2 n n = ?, 0 1 () 2 n n e = ? ?. 3.?闭?Re( )0 n a ,? 1 n n a = ? 2 1 n n a = ?,? 2 1 n n a = ?. ? 15 / 37 ?闭? 2222 i,(i)2i nnnnnnnnnn axy axyxyx y=+=+=+ ? 1 n n a = ? 2 1 n n a = ? ? 2 1111 ,() , nnnnnn nnnn xyxyx y = ? ?Re( )0 n a , ? 0 n x ? 2 limlim0 nn nn xx = ? n ?, 2 nn xx ? 2 1 n n x = ? 2 22222 2() nnnnnn axyxxy=+= ? 2 1 2 n n x = ? 22 1 () nn n xy = ? ? 2 1 n n a = ? 2 1 n n a = ?. 4.? 1 0 () nn n zz + = ? ? ? 11 0 ()1 n kkn n k szzz + = = ? 1,1 n zs令? . ? 1z = ? 1z ?, 1 0 () nn n zz + = ?. 5.? 0 (2)n n n Cz = ? z称代 ? z称3 ?. ? ? 1 lim n n n C C + = ? 1 2z ?. ? z称代 ?,? 1 2 ? z称3 ?, ? 1 1 ?,? 0 (2)n n n Cz = ? z称代 ? ? z称3 ? 6.? (令)?. (以) ? ?. ?闭 (令) ?,? ?,?. (以) ?,? ?. 7.? 0 n n n C z = ? R,? 0 n n n n C z b = ? ? ? ? 1 1 1 11 1 limlim n n n nn n n n C C b C CbR b b + + + = ? R R b = 8. ? ? ? ? ? ? ? 0 n n n a z = ? ? ? ? ? lim n n n a = ? (令)?0 ? ? 2 n n a z ? ? ? ? ?,lim 1 n n n n a z ?.? 1 R = . ? 0= ?, 1z ,? R = +. ?= +,? 0,z ?,? 2 n n a z ? ?,?.? 0R = 9.? (令) 0 (i)n p n z n = (以) 0 pn n nz = (3) 121 0 21 ( i) 2 nn n n z n = (4) (1) 0 i ( ) (1) nn n n z n + = ?闭 (?) 111 limlim()lim(1)1 (1)11 1 pp pp nnn n nnnn R = + = ? i1z ? 0 n n n C z = ? 1 z ? ? 1 zz ? 0 n n n C z = ? 0 n n n C z = ? ? 0 z ? ?,? 0 n n n C z = ? 0 zz ?. 令3.?ln(1 e ) z + ? 0z = ? ?,(? 4 z ?),?. ?闭? 1e ln(1 e )ln() e z z z + += ? (21)i(0, 1,.) k zkk=+= ? R = ? 0 ln(1e)ln2 z z = += 0 e1 ln(1 e) 1 e2 z z z z = += = + 0 22 e1 ln(1 e ) (1 e )2 z z z z = += = + 2 0 3 ee ln(1 e )0 (1 e ) zz z z z = + += + ? 18 / 37 2 (4) 0 43 e (1 4ee)1 ln(1 e ) (1 e )2 zzz z z z = + += + ? 24 23 111 ln(1 e )ln2., 22!24!2 z zzzR +=+= 令4.? 2 1 1z+ ? 12z ? ?,(? 4 (1)z ?) ? iz = ? 2 1 1z+ ? 2R = ? 2 11 ( ),(1) 12 f zf z = + 22 21 ( ),(1) (1)2 z fzf z = + 2 2 3 261 ( ),(1) (1)2 z fzf z + = + 3 24 2424 ( ),(1)0 (1) zz fzf z = + 24 (4)(4) 2 5 24240120 ( ),(1)0 (1) zz fzf z + = + ? ( )f z ? 1z = ? 24 2 11113 (1)(1)(1).,2 12244! zzzR z =+= + 令5.? ?. (令) 1 23z ? 0z = ? 1z = ? (以) 3 sin z ? 0z = ? (3) arctan z? 0z = ? (4) ( 1)(2) z zz+ ? 2z = ? (5) ln(1 ) z+ ? 0z = ? ? ?令? 0 1111123 () , 2 23323332 1 3 n n zz zz z = = = = 0 11111 2 (1) ,1 2322 12(1) 11 2(1)2 nn n zz zzzz = = = (以) 35 21 0 ( 1) sin. (21)!3!5! n n n zz zzz n + = =+ + 2 321 0 331 sin( 1), 4(21)! n nn n zzz n + = =代?b代? ? ( ) ()() 2222 1 R z zazb = + ?R(z)? ?z称ia? ib? ( ) ( ) () () ()() () ()() ()() () 22222222 ii 2222 2i Res, iRes, i 11 2i limilimi 11 2i 2i2i zaz b IR z aR z b z az b zazbzazb a bab ab ab a b =+ =+ + =+ = + ?4?. () 2 2 022 d x x xa + + ?a代? ? ()() 22 22 02222 1 dd 2 xx xx xaxa + = + ? ( ) () 2 2 22 z R z za = + ?z称ai ?R(z)? ? 23 / 37 ? ? () ( ) ( ) () ()() 2 2 22 22 22 i 0 i 1 d2iRes, iRes,i 2 i limlim ii 2 zaza x xR zaR za xa zz zaza a =+ + =+ + = (5) () 2 022 sin d x x xb x + + ?代?b代? ? ()()() i 222 222222 cossin eddid x xxx x xx xx xbxbxb + =+ + ? ( ) () 2 22 z R z zb = + ? ?R(z)?z称bi ? () ( ) () ii 2 22 i i ed2i Rese, i e 2i limei i2 z xz z b b x xR zb xb z zbb + = + = + () 2 22 sin de 2 b b b xx x x + = + ? () 2 022 sin de 44 e b b x x bb x xb + = + (6) 22 i e d x x xa + + ?a代 ? ( ) 22 1 R z za = + ?z称ai ? ? ( ) ii i 22 i eee d2i Rese , i2i lim2i i2 ie xza z a za xR za xaz aaa + = + 7. ? (令) () 2 0 sin2 d 1 x x xx + + ? ( ) () 2 1 1 R z zz = + ?R(z)? ?z称代?r?化r? ? 叶R,与-R, -r成, 叶r,与r, R成? ?i, ? () ( ) () 2 22 i2 0 1e1e ImdIm 2i Res,ilimd 2211 r r xiz c IxR zz zzxx + = + ? () 2 0 2 ed limi 1 r iz cr z zz = + ? ? () () 22 2 1e1e Im 2i limiIm 2ii1 e 2222 z i i z I z z i =+= += + ? (以) 2 1 d 2i z T a z z ? ?切?Rez称化, 化代, 代积a积令 ? ? ? ? ?z称化+iy (- ? 积 y 积+ ? ) ? ? ? ( ) ln 22 e zza a f z zz =?() ln 22 e i ca f cy cy += + ? () ln 22 e idd ca f cyyy cy + += + ? ? ? ? ? ? ? ( ) i i d c c f zz + ? ( )( )( ) ii ii dlimdlimd cc ccABRRR R f zzf zzf zz + + = ?AB?化-iR?化+iR? ? f(z)?-R?x?化?-R?y?R? ? ?f(z)?z称代? ( ) ( ) () 2 0 Res,0limln z f zzf za = ? ( )( )( )( ) dddd2i ln ABBEEFFA f zzfzzfzzfzza+= ? ( ) () () () ilnlnlnln 22222 eeee dddd0 i xR ax aa CC a RCC R B ECRR f z zxxxC R x RRR x R + + = + + + ? ? ? 令. ? 1 w z =?. (令) 22 xyax+= (0a ?) ? 2222 11 i= +i i xy wuv zxyxyxy = + 22 1xx u xyaxa = + , ? 1 w z =? 22 xyax+=? 1 u a =. ? 24 / 37 (以) .ykx=(k?) ? 2222 1 i xy w zxyxy = + 222222 xykx uv xyxyxy = = + vku= ? 1 w z =?ykx=?vku= . 以. ? ?令?Im( )0,(1i)zwz=+? ? (1i) (i )()i( + )wxyxyx y=+=+ ,.20.uxy vxyuvy=+= . ?(1i)wz=+?Im( )0,z ?Im( )Re( )ww. (以) Re(z)代. 代积Im(z)积令, i w z =. ?z称x+iy, x代, 代积y积令. 222222 iii(i ) i xyyx w zxiyxyxyxy =+ + Re(w)代. Im(w)代. ?w称u+iv, ? 2222 , uv yx uvuv = + ? 代积y积令,? 22 22 11 01,() 22 u uv uv + ? i w z =? Re(z)代, 代积Im(z)积令.? Re(w)代,Im(w)代, 1 2 1 2 w (? 1 2 ,代? 1 2 ?) 3. ?w称z 以? z称i ? ?w称z 以? ?z称i?w ?. ? w 称以z,? w (i)称以i, | w |称以, ? arg w 称 2 . ?, ? i ?w? ?-令?.?. ? 4. ? ?w称z 以? z? ? ? ?w称z 以? z称代 ? ?z称代 ?. 5. ? 代积x积令 ? wz=+?. 6. ?1?. ? azb w czd + = + (a北-b化代)? 11 .? 1 ab bacd cd + =+ + ? (1)a zcd w czd + = + , ? (1)(1) 1 a zc z w czd + + = + , ?11?22 ac ad cd + = + . ? 1 1(1)(1) d c d c dc wzz czdz + + =+=+ + ? d q c =? 1(1)(1) / ()(1)(1)1 1(1)(1) / ()2(1)(1)1 wzqzqzqz a wzqzqzqz + = + ?a?. ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 11 11 wz a wz + = ? a?. ? 25 / 37 7. ? azb w czd + = + ?|z|称令 ? ? ? azb w czd + = + ?|z|称令 ? d z c = ?w= . ? d z c = ? ? ? ? ? ? |z|称令, ? ? 1 d c =,|化|称|北|. ?a北-b化代,?|化|称|北|. 8. ? 1 1 z w z = + ?. (令) Re( )0z =问 (以) |z|称以问 (3) Im(z)代. ?(令) Re(z)称代 ?z称iy?. 22 222 i1(1i )12 i i1111 yyyy w yyyy + =+ + ? ? ? ? ? ? ? 2 2 1 1 y u y + = + ? 2 2 1 y v y = + , y 代 ? Im(w)代. 9. ? Re(z)代 ? |w|积令 ?. ?z代?w称代?z代? ? 0 z?w= ? ? 0 0 zz wk zz = ?k? ?. ? 0 0 zz wk zz = ,?z?|w|称令, ?z称代? i0 0 | 1e() zz wkkk zz = = R ? 0 0 0 e(Re()0) i zz wz zz = . 令代. ?e 1 i z w z = ?| 1z 代, ?|w|积令 ? ?w称k z z (Im()代). (令) ?f(i)称代 ?称i? arg(i)0 f =,? i 2 2i ( )e (i) fz z = + , i() 2 1 (i)e0 2 f =? 2 =? i i i z w z = + . (以) ?f(令)称令,?k称 1 1 ?f(i)称 1 5 ,? k称 i 5(i) ? 3 +( 52i) ( 52i)3 z w z = + . 令以. ?|z|积令 ?|w|积令 ? ? 26 / 37 w称f(z)? (令) f( 1 2)称代, f(-令)称令. (以) f( 1 2)称代, 1 2 arg( ) 2 f =, (3) f(a)称a, arg( )fa=. ?|z|积令 ?|w|积令 ? ? i e 1 z w z = , |积令. (令) ?f( 1 2)称代? 1 2 =.?f(-令)称令? 1 iii 2 1 2 1 ee ( 1)1e1 1 = = = + . ? 1 2 2 21 1 12 z zz w z = = . (以) ?f( 1 2)称代? 1 2 =? i 2 54 e (2) z w z = i 11 22 4 ( )earg( ) 32 ff =? ? 2 1 i 2 2 21 e ()i 12 z zz w z = . (3) ?=( ) z? ?z称a0=?arg( )a=, ?|z|积令 ?|积令. ? i =( ) = e 1 za z a z ?w称g()?称代w称a, arg(0)0 g =,? |积令 ?|w|积令. ? ? ? ? ? ?=( )w, ? ? |w|积令? ? |积令,w称a?称代? arg( )arg(1/(0)0wg=,? =( ) = 1 wa w a w . ?w?. i = e 11 waza a wa z . 令3. ? 代?令?i ? ? 代?以?令+i ? ?. ? 0 2 w w ? 1 i0 1 i2 + + 称 0 2 z z ? i0 i 1 2 w w . 1 i2 1 i + + 称 1 z z . i1 i 4z (i 1)(1 i) w z = + . 令4. ? 以积|z|积5 ? 4积|w|积令代 ?f(5)称-4 ?. ? z称5,-5,-以,以 ? w称-4,4,令代,-令代,? ? 25 25 + ? 25 25 + 称 104 104 + ? 104 104 + ?w称f(z)? 5 5 z z + ? 25 25 + 称 4 4 w w + ? 104 105 + ? 4 4 w w + 称 5 5 z z + 20 w z = . ?w称f(z)?|z|称以 ?|w|称令代?z称5 ?w称-4.?|z|以 ? |w|积令代.?w称f(z)?|z|称5 ?|w|称4?z称以 ? w称-令代?|z|积5 ?|w|4? ?. 令5. ? ? 2 wz=?z? ? ? ? ? ? 2 2 11 24 xy += ?w? ?. 令6. ?w称e z?. (令) ? Re(z)称叶令,Im(z)称叶以问 (以) ?Im( ),02z令, 0argw令 ?|w令|积令,? ?. 1 2 1 1 i 1 w w w + = ?|w令|积令 ? Im(w以)代, ? 2 32 ww=? ? ? 3 3 1 1 w w w = + ? ?与-令,令成?. ? 22 1 1 21 1 32 222 1 32 1 1 1 11 1i1 1111 11 () 112 11 i11 11 z z z z w www wz wwz w w + =+ + + + . 令8. ?Re( )0z 代? 1 2 1 1 ln 1 w w w + = ? (-,-令成? 32 iww=? ? ? 43 lnww=? ? 代积Im(w4)积,Re(w4)代? 4 2iww=? e1 ln e1 z z w + = . 令9. ?Im(z)积令?|z|积令? ? Im(w)代 ?. ?. 以代. ?coswz=? 代积Re(z)积,Im(z)代 ? ?. ? ? 1 cos() 2 iziz wzee=+ ? w令称iz ? 1 2 eww =? 32 2 11 () 2 ww w =+ ?coswz=? ? 令 ? w令称iz? ? ? ? ? ? 2 ? ? ? 代积Im(w令)积,Re(w令)积代. ?以? 1 2 eww =? |w以|积令,Im(w以)代. ?3? 2 2 11 () 2 ww w =+? Im(w)积代. ? 令.?f(t)?f(t) ? + = 0 dsin)()(tbtf ?( ) + = 0 tdtsin 2 )(tfb ?f(t)? + = 0 cos)()(tdwatf ? 28 / 37 ? + = 0 2 tdtcf(t)( osa ? ? dGtf ti + =e )( 2 1 )(?)(G?f(t) ? ( )( )( ) (cossin) i t Gf t edtf ttit dt + = ( ) cos( ) sinf ttdtif ttdt + = ?f(t)?tcosf(t)? + =0tdtcosf(t) tsinf(t)? + + = 0 .sinf(t)2tdtsinf(t)tdt ?.sinf(t)2)( 0 tdtiG= + ? )()(GG=? dtitGdeGtf ti )sin(cos)( 2 1 )( 2 1 )(+= + + 称 0 1 ( )sind( ) sind 2 i GitGt + = ? f(t)? 00 2 ( )b( ) sind .b( )=( ) sindt. f ttf tt + = ? ? f(t)? 0 ( )( ) cosdf tat + = .? 0 2 ( )( ) cos af ttdt + = 以.?( )f t = 2 1 , 0,1 t t t .?( )a? ?. ?闭 1 2 001 11 22 00 1 2 0 1 2 0 1 1 20 0 222 ( )( ) cosdcosd0 cosd 22 1 cosdd(sin) 1 22 sinsin2 d 0 2 sin4 (cos) 2sin4 coscos 2sin4co af tt ttt tt t tt ttt tttt t t dt tttdt + =+ = = =+ =+ =+ 23 s4sin 3.? sin ,6 ( ) 0,6 t t f t t = ?. ?闭 6 6 6 6 6 0 2 ( )( )dsind sin(cossin)d 2sinsind sin6 (1) i ti t F ff tett et ttitt itt t i + = = = = 4.? (1) ( ) t f te= ? | |(| |) 0 (1)(1) 2 0 F f ( )( )ddd 2 dd 1 i tti tti t titi f t eteetet etet + + + + = =+= + (以) 2 ( ) t f tt e= ? 2 2222 / 4 F.()( 2 )2. tttt eeeett e = = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2 2 4 ( )F() 2 t Gt ee i = (3) 2 sin ( ) 1 t f t t = ? ? 29 / 37 2 2 22 0 22 00 sin ( )F( )( )d 1 sin (cossin)d 1 1cos( )cos() sin sin 2 d2d 11 cos(+ )cos(- ) dd() 11 sin,| 2 0,| i t t Gfet t t titt t tt tt itit tt tt itit tt i + + + + = = + = = = = ? ? ?| . (4) 4 1 ( ) 1 f t t = + ?闭 444 44 0 1cossin ( )ddd 111 coscos 2dd 11 i t tt Gettit ttt tt tt tt + + = + = + ? 4 1 R(z)= 1z+ ?R(z)? ? 22 (1),( 1) 22 ii+ +. 22 R()d 2R( ),(1 ) 2R( ),( 1 ) 22 i ti zi z t e ti Resz eii Resz ei + =+ + + ?. | |/ 2 44 cos1| dRed (cossin) 11222 2 i t te tte tt + =+ + (5) 4 ( ) 1 t f t t = + ?闭 4 44 4 ( )d 1 sin cosdd 11 sin d 1 i t t Get t ttt t tit tt tt it t + + + = + = + = + ?(4).?,? 4 R( )= 1 z z z+ ? 44 | |/ 2 sin d()Im(d ) 11 sin 22 i t ttt e itit tt i e + = + = . 5.?(t)? Im( ) t代 ?,?u称at.? 11 F ()( )( )d u ia f atf ueuF aaa + = ?a积代 ?,?u称at,? 1 F ()( )F()f at aa = . ?. 9.?() ();FF f =? ()()()Ff t = F. ? ()( )()( ) ( ) () ( ) ( ) () () () ede d eded ed. i ti u iiu u it Fftfuf tut fufu uu ftF t + + + = = = 令代.?() ()FF f =? ( )()()() 000 1 cos 2 F ftFF t =+ ? ( )()()() 000 1 sin. 2 F ftFF t =+ ? ( ) ( ) ( )( ) ()() 00 00 0 00 e+e cos 2 1 ee 2 22 1 2 itit itit F ftF tft FF ff tt FF = =+ =+ ? ( ) ( ) ( )( ) ()() 00 00 0 00 ee sin 2 1 ee 2 1 2 itit itit F ftF ft t i FFff tt i FF i = = =+ 令令.? ( )( ) 0,0sin , 0t 2 00e , t tt fg tt t ? 0 0 e( )() s t LfsF ss= ?闭 00 00 0 ()() 0 00 e( )e( ) e ( ) e( ) e() s ts tst ss ts st Lfsf tdt f tdtf tdtF ss = = 7