《復(fù)變函數(shù)與積分變換》(西安交大_第四版)課后答案

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1、 1 1 i 2 3 1 + 2 i 1 3i i 1 3 ( ) ( ) 2i 5i 2 4i 3 + 4 i 4i i 21 8 + 1 () () () 2i 3 13 1 2i 3 2i 3 2i 3 2i 3 1 = + = + 13 3 = + i 2 3 1 Re 13 2 2i 3 1 Im = + , () 2i 3 13 1 2i 3 1 + = + 13 13 13 3 13 3 2i 3 1 2 2 = + = + k 2 i 2 3 1 arg i 2 3 1 Arg + + = + , 2 , 1 , 0 , 2 3 2 arctan = + = k k 2 ()

2、( ) () () i, 2 5 2 3 3i 3 2 1 i i) (1 i 1 i 1 3i i i i i 1 3i i 1 = + = + + = , 2 3 i 1 3i i 1 Re = 2 5 i 1 3i i 1 Im = 2 5 i 2 3 i 1 3i i 1 + = 2 34 2 5 2 3 i 1 3i i 1 2 2 = + = k 2 i 1 i 3 i 1 arg i 1 i 3 i 1 Arg + = , , , = , + = 2 1 0 2 3 5 arctan k k . 3 () ( ) () () () () ( ) ( ) ( ) 4 2i 7i 2

3、6 2i 2i 2i 5i 2 4i 3 2i 5i 2 4i 3 = + = + 13i 2 7 2 26i 7 = = () ( ) 2 7 2i 5i 2 4i 3 Re = + () ( ) 13 2i 5i 2 4i 3 Im = + 1 課后答案網(wǎng) () ( ) l3i 2 7 2i 5i 2 4i 3 + = + () ( ) 2 29 5 2i 5i 2 4i 3 = + () ( ) () ( ) k k 2 7 26 arctan 2 2 i 2 i 5 2 i 4 3 arg i 2 i 5 2 i 4 3 Arg + = + + = + () , 2 , 1 , 0 ,

4、 1 2 7 26 arctan = + = k k . 4 ( ) ( ) () () i i 1 4 1 i i i 4 i i 4i i 10 4 10 2 4 2 21 8 + = + = + 3i 1 i 4i 1 = + = 3 i 4i i Im 1, i 4i i Re 21 8 21 8 = + = + 3i 1 i 4i i 21 8 + = + 10 | i 4i i | 21 8 = + ( ) ( ) ( ) 2k 3i 1 arg 2k i 4i i arg i 4i i Arg 21 8 21 8 + = + + = + = . 2, 1, 0, k 2k arc

5、tan3 = + 2 ( ) i 1 3i 5 3 y i 1 x + = + + + x, y ( ) ( ) ( ) () () 3i 5 3i 5 3i 5 3 y i 1 x 3i 5 3 y i 1 x + + + = + + + ( ) ( ) ( )() 34 3 y 5 1 x 3 i 3 y 3 1 x 5 + + + + + = () i 1 18 5y 3x i 4 3y 5x 34 1 + = + + + = = + = + 34 18 5 3 34 4 3 5 y x y x = + = + 52 5 3 38 3 5 y x y x 11 , 1 = = y x 3

6、 i -i=i -1 = i 4 2 1) | | 11 6) Re( ) ( ), Im( ) ( ) 22 i zz z zz zzz = z = += 2 課后答案網(wǎng) i zxy = + 5 z 2 | zz = 2 2 2 z i zxy =+ 2 | zz = 22 2 2i x yx yx + =+y 0 2222 , xyxyx y =+ = z 6 n a 1 | | z | | a z n + |a| |a| |z| a z n n + + + 1 n a e z arg i = () |a| e a |a|e e a| z a a n n a n + = + = + = +

7、 | 1 1 arg i arg i arg i | | 1 a + 8 1 i 2 -1 3 1+ 3 i 4 ( ) 0 isin cos 1 + 5 i 1 2i + 6 ( ) () 3 2 isin3 cos3 isin5 cos5 + 1 2 i e 2 isin 2 cos i = + = 2 i e isin cos 1 = + = 3 3 i 2e 3 isin 3 cos 2 2 3 i 2 1 2 3 i 1 = + = + = + 4 2 1 cos isin 2sin i2sin cos 2sin sin icos 222222 +=+ = + ) (0 , e 2

8、2sin 2 isin 2 cos 2 2sin 2 i = + = 5 () = = = + 2 1 i 2 1 2 i 1 i 1 2i 2 1 i 1 2i = 4 isin 4 cos 2 = 4 i e 2 6 () () ()() 2 23 i5 i3 i10 i9 i19 3 cos5 isin5 e/ e e/ ee cos3 isin3 + = = 3 課后答案網(wǎng) isin19 cos19 + = 9 1 11 11 , ; x xa yyb = + = + 2 11 11 cos sin , sin cos . xx y yx y = =+ 11 i Aab = + 11

9、i zxy 1 = + i zxy = + 1 2 1 zzA =+ i 11 (cos i sin ) e zz z =+= 10 -i z e z z Arg i | | = () = | | = 2 Arg i 2 i Arg i z z |z|e e e z i z 2 11 222 12 12 1 2 | 2 ( | | zz zz z z += + 2 | ) 22 12 12 12121212 11 22 22 12 |() () () ( 2( ) 2(| | | | ) zz zz zzzz zzzz zz zz zz +=+ =+ =+ ) 12 1 () () () Pz

10、 Rz Qz = i X Y + X Y x y 2 () R z 1 () i RzXY = 3 i ab + 1 01 1 0 nn nn az az a z a + + += i ab 1 () ()() R e ()() ) I m ()() ) () () (,) (,) ()() Pz PzQz PzQz PzQz Rz Qz qxy qxy QzQz = = + ; 2 () () () () i i () () () Pz Pz Pz RzX Qz Qz Qz Y X Y = = = =+= ; 3 ( ) 1 01 1 nn nn Pz az az az a = + + 4

11、課后答案網(wǎng) () z P z a z a z a a n n = + + + + = 2 2 1 0 13 it e z = 1 nt z z n n cos 2 1 = + 2 nt z z n n sin i 2 1 = 1 nt e e e e z z n n sin 2 1 int int int int = + = + = + 2 nt e e e e z z n n sin i 2 1 int int int int = = = 14 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 3

12、2 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 ()

13、() 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 15 (1 i ) (1 i ) nn + = n 5 課后答案網(wǎng) i /4 i /4 i /4 i /4 (2 e ) (2 e ), e e nn nn = sin 0 4 n =

14、 4, 0 ,1 ,2 , nk k = 16 1 0 8 3 = + z 2 0 8 = + y y 1 () () 1 i12 3 3 82 k ze + = = k=0,1,2 , 3 i 1 + , 2 3 i 1 2 0 8 3 = + i 3 1 1 + = 2 2 = i 3 1 3 = ( ) x C x C e e C y x x 3 sin 3 cos 3 2 2 1 + + = 17 z 11 1 , , zz z zz z o x y z -z z z 1 z 1 z 1 z 18 1 z 2 z 3 2 1 , , z z z 1 () 2 1 2 1 z z z +

15、 = 2 ( ) 2 1 1 z z z + = 3 () 3 2 1 3 1 z z z z + + = 1,2,3 i = + = ,k y x z k k k 1 2 i 2 2 1 2 1 y y x x z + + + = z 1 z 2 z 6 課后答案網(wǎng) 2 ( ) ( ) 1 2 2 1 2 2 i y y y x x x z + = 1 z 2 z | z |z | z |z 1 2 1 = 3 () ( 3 2 1 3 2 1 3 i 3 1 y y y x x x z + + + + + = ) z 3 2 1 z z z 19 123 , zzz 0 3 2 1 = +

16、 + z z z , 1 3 2 1 = = = z z z z 1 z 2 z 3 1 = z 1 3 2 1 = = = z z z 3 2 1 z z z 2 33 1 zz = 3 z () () 2 1 2 3 2 2 1 1 2 1 2 1 z z z z z z z z z z z z + + + = + + = 2 1 2 1 2 z z z z + + = 1 2 1 2 1 = + z z z z ) ( ) )( ( 1 2 2 1 2 2 1 1 2 1 2 1 2 2 1 z z z z z z z z z z z z z z + + = = ( ) 3 2 2 1

17、2 1 = + = z z z z 3 2 1 = z z 3 3 2 3 1 = = z z z z 3 2 1 z z z 1 = z 20 z 1 z 2 z 3 3 2 3 1 1 3 1 2 z z z z z z z z = 3 2 1 3 1 2 z z z z z z = = ) arg( ) arg( ) arg( ) arg( 3 2 3 1 1 3 1 2 z z z z z z z z = 2 3 1 3 1 2 z z z z z z = () ( ) () () 1 2 3 2 3 2 1 3 3 1 1 2 1 3 1 2 z z z z z z z z z z

18、z z z z z z = + + = 1 2 3 3 1 2 z z z z z z = 3 2 1 z z z 3 2 1 3 1 2 z z z z z z = = 7 課后答案網(wǎng) 21 z 1 |5 2 | z=6 1 | i 2 | + z 3 Re( 2) 1 z + = 4 () 3 i Re = z 5 | i | | i | = + z z 6 4 | 1 | | 3 | = + + + z z 7 Im( ) 2 z 8 1 2 3 z z 9 0a r g z z 4 1 z 3 1 Re 0 z 4 23 z 5 3 1 + z z 6 1a r g 1 z z x y

19、 5 O 1 16 ) 1 ( 2 2 = + y z 3 0 1 Re + x z z y -1 D O x x = -1 6 1a r g 1 z D 1/2 y x 22 4 4 15 xy1 = 10 (2 2 i) ( i) 4 zz z z + x y o 2 -1 12 課后答案網(wǎng) 9 23 z 22 (2 ) 1 xy += C z a z a = + a C C By Ax = + ) ( i 2 1 Im ), ( 2 1 Re z z z y z z z x = = + = = C z B A z B A = + ) i ( 2 1 ) i ( 2 1 ) i ( 2 1

20、 B A a = + ( 2 1 a = ) iB A C z a z a = + 24 0, ( zz z z c ) c + += 22 () 2 () 0 zazaR z za za za aR +=+= c a aR = 25 t 1 2 t b t a z sin i cos + = t z i) 1 ( + = 3 t t z i + = 4 2 2 i t t z + = 5 6 7 ze ab = + 1 t ch i sh zatbt =+ ii tt za eb e =+ ,( i ) = = + = + = t t y t x t y x z , ) i 1 ( i x

21、y = 2 sin sin i cos i = + = + = b y t b t a y x z 2 0 , cos = t t t a x 1 2 2 b 2 2 = + y a x 3 = y 1 1 = + = + = t t x t t y x z i i = xy = = + = + = 2 2 2 2 1 i i t y t x t t y x z 4 1 = xy 13 課后答案網(wǎng) 22 22 ch ch i sh 1 sh xat xy zatbt ybtab = = + = = 5 22 22 1, ()() xy ab ab += + 6 2 arctan 22 ay b

22、x xye += 7 26 z w 1 = z w () iv u w iy x z + = + = , 1 2 6 2 2 = + y x x y = 3 ( ) 1 1 2 2 = + y x 1 x = 4 2 2 2 2 1 1 y x y i y x x iy x z w + + = + = = 2 2 2 2 , y x y v y x x u + = + = 1 ) ( 22 22 11 xy uv x + += = = 2 w 2 22 22 4 xy y + + () v y x y x + + 2 2 2 2 y y y x x u = = = + = 2 2 x = w

23、3 1 2 1 , u = 1 y + 2 1 y v y = 22 2 1 1 uv u y += + + 22 2 11 22 uv += 4 w () 2 1 2 1 1 2 2 2 2 2 2 = + = + = + y x x x y x y x 2 1 = u 27 w v 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

24、 3 z = 2 6 sin i 6 cos 2 2 1 i 3 i e = + = + = + 14 課后答案網(wǎng) 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 0 0| | zz |() |1 fz A 0 0| | zz |() |() |() |1| fzfzAAfzAA A = + + + 31 1 () , ( 0 ) 2i zz fz z zz = 0 z () f z 2 x 2 12 () 2i zz x y fz zz y = + 32 )

25、 arg ( arg z z z z z f arg ) ( = f(0) f(z) z=0 0 0 z 0 ) 0 0 ( = x x z = x x x arctan lim 0 + = y y z y y x z z arctan lim arg lim 0 0 arg 0 + x x 0 0 z arg argz z z lim z z 15 課后答案網(wǎng) 1 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 + =

26、+ + = 2 2 00 11 11 1 l i m l i m () zz zzz zzz z z z + = = + 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

27、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 3 () f z 1 5 (1 ) z 2

28、 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 課后答案網(wǎng) 2 4 () 2 () ad bc fz cz d = + ( ) z f /( 0 ) zdc c = 5 1 0 z 0 z D D 2 2 6 1 () z f 0 z ( ) 0 z f 2 () 0

29、 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 =

30、 () () 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 7 () 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 課后答案網(wǎng) 3 () 2 2 | | v u x v v x u u z f x + + = () 2 2 | | v u y v v y u u z

31、 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 = + + = + + + + = 9 - = v r r u 1 = u r r

32、 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 10 () iv u z f + = D () z f 1 () z f

33、 2 () z f D 3 () | | z f D 4 () z f arg D 5 c bv au = + a b c 課后答案網(wǎng) 4 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

34、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 , 0 0 , 0 1 1 1 1 v u C v u C u C () z f arg x y 課后答案網(wǎng) 5 0 1 1 2 2 2 2 =