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1.1 z 2 | zz = 2 2 2 z i zxy =+ 2 | zz = 22 2 2i x yx yx + =+y 0 2222 , xyxyx y =+ = z 1 1.2 1 ( 5 i 3 ) 2 ( ) 6 i 1 + 3 6 1 4 () 3 1 i 1 1 () () 6 / 5 i 5 6 / i 5 5 32 2 2 i 2 3 2 i 3 = = = e e 5 5 32 cos isin 16 3 16i 66 =+= 2 () () 6 6 6 i/ 4 3i / 2 1i 1i 2 2 e 8 e 8 i 22 += + = = = 3 () () 1 i 21 / 6 i +2 6 6 1 e e , 0,1,2,3,4,5 k k k + = = = 6 1 6 , 2 i 2 3 e /6 i + = i e /2 i = 2 i 2 3 ei /6 5 i + = 2 i 2 3 e /6 i7 = i 2 3 i = / e 2 i 2 3 4 11 i = / e 4 () () 0,1,2 = , = = 2 2 1 2 = 1 3 + 3 1 / 3 1 3 1 k e e k 2 4 i 6 4 i 2 2 i i 3 () 1/3 1i , 12 7 sin i 12 7 cos 2 2 , 12 sin i 12 cos 2 2 6 12 / 7 i 6 6 2 / i 6 + = = e e + = 4 5 sin i 4 5 cos 2 2 6 4 / 5 i 6 e 2 1.3 1 0 8 3 = + z 1 () () 1 i12 3 3 82 k ze + = = k=0,1,2 , 3 i 1 + , 2 3 i 1 1.4 z 1 |5 2 | z=6 1 | i 2 | + z 3 Im( ) 2 z 4 0a r g z z 3 1 Re 0 z x y 5 O 1 16 ) 1 ( 2 2 = + y z 3 0 1 Re z x = 0 x = 1 x O y 2 3 Ox y 4 23 z 2 3 3 1.6 3 z w = 1 i 1 = z i 1 2 + = z i 3 3 + = z 2 w 3 arg 0 z w i re z = 3 3 3 i e r z = = 4 2 2 i 1 4 sin i 4 cos 2 i 1 , i z e z = + = + = = = i 2 e 1 6 i 3 2 2 3 z = 2 6 sin i 6 cos 2 2 1 i 3 i e = + = + = + 4 w i 3 / 3 i 1 = = e w 2 i 2 2 1 3 2 3 i 2 1 2 2 2 4 2 + = + = = e w i 8 2 2 i 3 3 = = e w 2 w arg 0 1.7 1 () , ( 0 ) 2i zz fz z zz = 0 z () f z 2 x 2 12 () 2i zz x y fz zz y = + 1.8 ) arg ( arg 0 0| | zz |() |1 fz A 0 0| | zz |() |() |() |1| fzfzAAfzAA A = + + + 5 6 2.1 1 2 11 1)( ) , ( ) 2 nn zn zn zz = 1 122 1 1 00 () () l i m l i m ( ) nn nn n n n n zz zzz zn z C z z z n z z + =+ + = 2 2 00 11 11 1 l i m l i m () zz zzz zzz z z z + = = + 2.2 1 () y x z f i 2 = 2 33 () 2 3i fzxy =+ 3 () y x xy z f 2 2 i + = 4 () s i nc h i c o ss h fzx yx y = + 1 1 , 0 , 0 , 2 = = = = y v x v y u x x u z 2 1 = x u,v C-R () y x v u z f i i = + = 2 1 = x z 2 2 6 u x x = 0 u y = 0 v x = 2 9 v y y = z 22 23,230 xyxy = = u,v C-R () 33 i23i fzuvxy =+= + 230 xy = z 3 2 y x u = xy y u 2 = xy x v 2 = 2 x y v = z z=0 u,v C-R ( ) y x xy z f 2 2 i + = 0 = z z 4 cos ch u x y x = sin sh u x y y = sin sh v x y x = cos ch v x y y = z u,v C-R () s i nc h i c o ss h fzx yx y = + z z 2.3 () f z 1 5 (1 ) z 2 3 2i zz + 3 2 1 1 z 4 (, 0 ) az b cd cz d + + 1 () 4 5( 1) fz z = ( ) z f z 2 () i 2 3 2 + = z z f () z f z 3 () () () () 2 2 2 2 1 1 2 1 2 + = = z z z z z z f () z f 1 = z z 1 = z ( ) z f 7 4 () 2 () ad bc fz cz d = + ( ) z f /( 0 ) zdc c = 1 1 0 z 0 z D D 2 2 2 1 () z f 0 z ( ) 0 z f 2 () 0 z f () z f 0 z 3 0 z () z f ( ) z f 0 z 4 0 z () z f () gz 0 z ( ) () fzgz + ( )/() f zgz 5 (,) uxy (,) vxy () i fzuv = + 6 () i fzuv =+ u () f z D v () f z D 1 () 2 2 2 | | y x z z f + = = z z=0 2 () 2 | | z z f = z=0 z=0 3 00 () () fz z z fz 4 () s i nc h,() i c o ss h fzxy g zxy = (/ 2 , 0 ) z = () () fzgz + 5 () xy x z z z f i Re 2 + = = z=0 C-R () z f z=0 6 u C-R v () f z D 2.5 () v u z f i + = z () () () 2 2 2 | | | | | | z f z f y z f x = + () 2 2 | | v u z f + = () f z D () f z 0 z () f z D () f z 0 z () f z 0 z 2.4 8 () 2 2 | | v u x v v x u u z f x + + = () 2 2 | | v u y v v y u u z f y + + = () v u z f i + = y v x u = x v y u = () () + + = + 2 2 2 2 2 2 2 2 1 | | | | x v u x u u v u z f y z f x + + + + x u x v uv x v x u uv x v v x u v 2 2 2 2 2 2 () () () 2 2 2 2 2 2 2 2 2 2 2 2 2 2 | | | | 1 1 z f z f v u v u x v x u v x v x u u v u = + + = + + + + = 2.7 - = v r r u 1 = u r r v 1 sin , cos r y r x = = v u, C-R sin cos y u x u r u + = () r u r r x u r y u r y v r x v v = + = + = cos sin cos sin = v r r u 1 () cos sin r y u r x u u + = sin cos sin cos x u y u y v x v r v + = + = = = u r r x u r y u r 1 sin cos 1 = v r r u 1 = u r r v 1 2.8 () iv u z f + = D () z f 1 () z f 2 () z f D 3 () | | z f D 4 () z f arg D 5 c bv au = + a b c 9 1 () z f 0 = v ( ) z f D C-R 0 = = y v x u 0 = = x v y u u D ( ) C u = ( ) C iv u z f = + = 2 () iv u iv u z f = + = D ( ) y v y v x u = = ( ) x u x v y u = = 1 () iv u z f + = D y v x u = x v y u = 2 1 2 0 = = = = vy v x v y u x u v u, D ( ) 2 1 2 2 1 1 , , C C C u C u = = ( ) C iC C iv u z f = + = + = 2 1 3 () | | z f D 1 C 2 1 2 2 C v u = + x y = + = + 0 2 2 0 2 2 y v v y u u x v v x u u () z f D C-R x v y u y v x u = = , = + = 0 0 y u u x u v y u v x u u 0 = = y v x u 0 = = vy v x v v u, 2 1 , C v C u = = () C iC C iv u z f = + = + = 2 1 4 z arg D 1 C ( ) 0 z f ( ) 0 + = iv u z f () = 0 , 0 , arctan 0 , 0 , arctan 0 , arctan arg v u u v v u u v u u v z f 0 ， 由 高 阶 导 数 公式有 |z|=1 e z z n dz= |z|=1 e z (z 0) (n 1)+1 dz = 2 i (n 1)! (e z ) (n 1) | z=0 = 2 i (n 1)! . 3.7 令C : 是 一 条 以0 为 心 ， 以r 1 为 半 径 的 圆 周 ， 且 完 全 含 于|z| 1 2222 363,363 xy uxx yy uxx yy =+ = 22222 ( )i363i ( 363)3 ( 1) xy f z u u x xy y x xy y i z =+ = 3 () ( 1 ) i, fz iz cc = + 2 22 22 2 22 222 2 22 2 22 22 i1 ( ) i i ()()()( ) yx x y xy x y xy z fzv v xyx yx yz zz =+= + = = = + 11 1 () , ( 2 ) 0 () 2 fz c f fz zz = + = = 3 ( ) i 2 2i( 1) 2i( 1 i ) 2i( 1) xy fzu u y x x y z = = += 22 () i ( 1 ) , ( 2 ) i () i ( 1 ) fz z c f fz z = + = = 4 222222 i1 ( ) i i yx xyx y z fzv v x yxyxyz zz =+= + = = + () l n , fz zcc = + 3.11 () i fzuv =+ 1 i() f z 2 u v 3 2222 22 2 22 | ( )| | ( )| 4( ) 4 | ( ) | xx fz fz uv fz xy += + = 1 i() i fzvu = () i fzuv =+ , uv C R () xy vu = () yx vu = i() f z 2 () i fzuv =+ i() i fzvu = u v 3.12 u v v u u v 3.11 3 222222 22 222 2 |() | |() | () () fz fz uv xyx y += + 2222 22 2 22222 ( )2 ( ) 4( ) 4 | ( ) | x x y y xx yy xx yy xx uvuvu uuv vv uv fz =+ + =+= 1 u 1 () , u f ax by a b = + 2 y uf x = 1 22 , , , xx xy y ua fua fub f = 0 xx yy uu + = 0 f = 12 () f ca xb y c =+ 2 2 234 2 11 , 2 , , , xx x yy y yy y uf uff uf uf xxxxx = = + = = 0 xx yy uu += 2 12 2 1 2 0 ,a r c t a n yy y f ff cc xx x +=+ 16 4.1 n 1 1i 1i n n n + = 2 i (1 ) ; 1 n n n = + + 3 i/2 1 n n e n = 1 2 22 1i1 2 i 1i1 1 n nnn nnn + =+ + 2 2 12 lim 1, lim 0 11 nn nn nn 2 = + n lim 1 n n = 2 n (1 ) n n 3 i/2 111 cos i sin 22 n n nn e nnn = 11 lim cos 0, lim sin 0 22 nn nn nn = = n lim 0 n n = 4.2 1 1 i n n n = 2 1 (6+5i) 8 n n n = 3 2 cosi 2 n n n = 1 ic o s i s i n 22 n nn =+ 1 cos 2 n n n = 1 sin 2 n n n = 1 i n n n = i1 n nn = 1 i n n n = 17 2 (6+5i) 61 88 n n n = 1 61 8 n n = 1 (6+5i) 8 n n n = 3 cosi ch n =n ch lim 0 2 n n n 2 cosi 2 n n n = 4.3 1 1 () n p n z p n = 2 1 ! n n n n z n = 3 0 1) nn n iz = 4 1 ln i n n z n = 1 1/lim lim 1 n p n n nn Ran = = 18 2 1 1 1 (1 ) 1/ lim lim lim 0 1 n nn nnn nn aa n R aan + + + = + = ; 3 1/lim lim1/|1 i| 1/ 2 n n n n Ra =+ = 6 1/lim lim|lni | n n nn Ran = ; 4.4 z 1 3 1 1 z + 3 () 2 2 1 1 z + 6 5 2 sin 2 z e z 1 z z e 7 z 1 1 sin 1 1 | | , 1 1 1 3 2 + + = + z z z z z () + + + + = + n n z z z z z 3 9 6 3 3 1 1 1 1 1 | | z R=1 3 () + + + + = + n n z z z z z 1 1 1 1 3 2 1 | | z () + + + + = + n n z z z z 2 4 2 2 1 1 1 1 1 | | z + 2 1 1 z () 2 2 1 2 z z + = () + + = + = + 6 4 2 2 2 2 4 3 2 1 1 1 2 1 1 1 z z z z z z 1 | | z =1 R 2 2 4 4 2 sin z 2 sin z 2 1-cos 2z = (z-a)(z-b) 1 5 23 1, | 2! 3! z zz ez z =+ + + + | , 23 1 0 ,| | 1 , 1 n n z zzz z z z + = = = 12 13 23 100 1 0 ()() 11 2! 3! 2! 3! nn z nnn z n zz zz ez z + += = = + + = + , | | 1 z R=1 7 , 1 sin 1 cos 1 cos 1 sin 1 1 sin 1 1 sin z z z z z z z + = + = , 1 | | , 1 0 1 3 2 = + + + = = + z z z z z z z n n () () + + + + + + + = 3 3 2 3 2 ! 3 1 1 sin z z z z z z z z + + + = 3 2 6 5 z z z 1 | | z ( )() + + + + + + + = 4 3 2 2 3 2 ! 4 1 2 1 1 1 cos z z z z z z z z + = 3 2 2 1 1 z z 1 | | z + + + + + = 3 2 3 2 6 5 1 cos 2 1 1 1 sin 1 1 sin z z z z z z = () 1 | | , 1 sin 1 cos 6 5 1 sin 2 1 1 cos 1 cos 1 sin 3 2 + + + + z z z z R=1 4.5 Taylor 0 z 6 , | | , ! 3 ! 2 1 6 4 2 2 + + + + + = z z z z e z , | | , ! 5 ! 3 sin 10 6 2 2 + + + = z z z z z + + + + + + = ! 5 ! 3 . ! 3 ! 2 1 sin 10 6 2 6 4 2 2 2 z z z z z z z e z , | | , 3 6 4 2 + + + + = z z z z ; + = R 1 1 1 + z z 1 0 = z 2 () () 2 1 + + z z z 2 0 = z 3 2 1 z 1 0 = z 4 z 3 4 1 i 1 0 + = z 19 1 () () 2 1 1 1 2 1 2 1 1 1 1 1 + = + = + z z z z z z 1 | | , 1 1 1 3 2 + + = + z z z z z () + + + = + 1 1 2 2 1 1 2 1 2 1 1 2 1 1 1 n n z z z z z z () + + + = n n z z z 2 1 1 2 1 2 1 1 2 () () n n n n z 1 2 1 1 1 = = 2 | 1 | z R=2 2 () () 1 1 2 2 1 2 2 4 2 1 2 1 + + = + + = + + z z z z z z z () 4 2 1 1 4 1 2 4 1 2 1 + = + = + z z z 4 | 2 | , 4 2 4 2 1 4 1 2 + = z z z () 3 2 1 1 3 1 2 3 1 1 1 + = + = + z z z = 3 | 2 | , 3 2 3 2 1 3 1 2 + z z z () + = 2 2 2 2 2 2 1 2 2 1 4 2 z z ( ) + 2 2 3 2 3 2 1 3 1 z z = () ( ) ( )( ) = = 0 0 2 3 2 1 3 1 2 2 1 2 1 n n n n n n n n z z ( ) () ( ) () = + = + = 0 1 0 1 2 2 3 1 2 2 1 n n n n n n n n z z () () = + + = 0 1 1 2 2 3 1 2 1 1 n n n n n z 3 | 2 | z 3 = R 3 = z z 1 1 2 () () () + + + + + = + = 2 1 1 1 1 1 1 1 z z z z 1 | 1 | + z () () + + + + + + = 1 2 1 1 2 1 1 n z n z z () () = + + = 0 1 1 n n z n 1 | 1 | + z R=1 20 4 () i 3 3 i 1 3 4 1 3 4 1 + = z z () i 1 3 i 3 1 1 + = z () i 1 i 3 1 3 1 1 i 3 1 1 + = z () () + + + + + = 2 2 i 1 i 3 1 3 i 1 i 3 1 3 1 i 3 1 1 z z () 1 i 1 i 3 1 3 + z () () n n n n z z i 1 i 3 1 3 3 4 1 0 1 + = = + () 3 10 3 i 3 1 | i 1 | = + z 3 10 = R 2 | zR (,) R R () f z z () f z z () (0) ! n n f c n = () f z (,) R R () (0) n f 21 1 1 2 3 Taylor 0 z 0 z 1 =0 n n z 1 z 1 = z 2 (3) ( ) z z f = , Taylor 3 23 1 z zzz z = + 2 11 1 1 z zz z = + 0 11 zz zz += 2 11 1 zz + + 23 0 zzz + += z | |1 z 4.6 Laurent 1 () () 2 1 1 2 + z z 2 | | 1 z 2 () 2 1 1 z z 1 | 1 | 0 , 1 | | 0 z z 3 () () + | 2 | 1 , 1 | 1 | 0 , 2 1 1 z z z z 4 2 1 (i zz ) i 5 z 1 1 sin + | 1 | 0 z 6 (1 ) (2 ) ,3 | | 4,4 | | (3 ) (4 ) zz zz zz + 1 2 5 1 1 5 2 1 5 1 ) 2 )( 1 ( 1 2 2 2 + + + + = + z z z z z z 2 1 1 10 1 1 1 1 1 5 2 1 1 1 1 5 1 ) 2 )( 1 ( 1 2 2 2 2 2 z z z z z z z z + + = + 22 () () = = = = 00 0 2 1 2 2 2 2 10 1 1 1 5 2 1 1 1 5 1 nn n n n n n n z z z z z z () () = = + = + = 0 0 ) 1 ( 2 0 1 2 2 10 1 1 1 5 2 1 1 5 1 n n n n n n n n n z z z + + = 80 40 20 10 1 1 5 1 1 5 2 1 5 1 1 5 2 3 2 2 3 4 z z z z z z z 2 | | 1 z 2 1 | | 0 z () ( ) 2 2 2 1 1 1 1 + + + + + = n z z z z z z () ( ) + + + + + + = n z n z z z 1 3 2 1 1 2 () + + + + + + = 1 1 3 2 1 n z n z z () = + = 1 2 n n z n 1 | 1 | 0 z ()()()() () ( ) = = + = 0 2 2 2 1 1 1 1 1 1 1 1 1 1 1 n n n z z z z z z () ( ) n n n z 1 1 2 = = 3 1 | 1 | 0 z () 1 1 1 1 1 1 1 2 1 ) 2 )( 1 ( 1 = = z z z z z z () () = = = 0 1 1 1 1 1 1 1 1 n n z z z z () = = 1 1 n n z + | 2 | 1 z () 1 2 1 2 1 1 1 2 1 ) 2 )( 1 ( 1 + = = z z z z z z 2 1 1 1 2 1 2 z 1 + = z z () () n n n z z z 2 1 1 2 1 2 1 0 = = () () = + + + = 0 1 1 2 1 1 2 1 n n n z z () () = + = 1 2 1 1 2 1 n n n z z () () = = 2 1 2 1 1 n n n z 23 4 0| i |1 z
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