Елисеев - Введение в методы теории функций...
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В.И.ЕЛИСЕЕВ ВВЕДЕНИЕ В МЕТОДЫ ТЕОРИИ ФУНКЦИЙ ПРОСТРАНСТВЕННОГО КОМПЛЕКСНОГО ПЕРЕМЕННОГО Издание второе, дополненное, переработанное E-mail: [email protected] [email protected] http://www.maths.ru/ Москва, 1990 - 2003 г.
Transcript of Елисеев - Введение в методы теории функций...
..
, , E-mail: [email protected] [email protected] http://www.maths.ru/
, 1990 - 2003 .
1. 1. . 2. A. 5 11 11 11 13 14 18 19 26 31 35 38 38 43 44 47 49 49 51 53 53 56 61 62 62 67 77 78 80 90 100 100 105 113 115 129 129 131 135 147 161 161
C. (v) D. e E. ln() F. G. H. , 3. 4. . 5. 6. . 7. (. ). 8.
= vn = n v B. = 1 / v
2. , 1. Z- 2.
170 170 172
2
3. 1. 2. 3. 4. -
179 179 183 187 188 188 189 193 196 199
4. 1. 2. 3. . . 4.
205 205 209 210 213 219 219 227 227 229 236 238 243 244 244 248 253 256
5. . 1. 2. 3. 4. 1. 2. . 3. 5. . . . . . . . .
6. . . .1. . . 2. . . . 3. . 4. . . 5. . .
258 258
265 290 293 301 3
7. . .1. , . 2. . . . 3. . 4. . 5. . 6. . 7. - . . , . 8. . 9. . .
311 311 313 315 316 318 319 322 330 341
8. . .1. . . . 2. . 3. S, C, B, t . . 4. , , u, d. 5. . . 6. . 7. . 8. 9. . , , , . -. 10. , . 8.2. , , , . 8.3.
345
345 349 361 362 366 368 375 381 397
415
418 422 432 432 436 440 457 500
9. 1. 2. . 3. .
10.
4
, (1789-1857 .). , . , . . , . , , , . , , . , , . , , . . . . , , , , . 140 . N . - , . . , , . . , . . , , . . , . , . . , . . , -. , , . , , 5
. , , . . . . , . . , . , , , . , , . , , , . N- ( N- , ) . . N- N- , N-. , ( ), , . , . . , . 4- . . . , , , . .
6
, , . - , , . , . , , , , , . . , , , . , , . . . , . , , , . , , , , . . . , , , . . , -. . . , , . 4- 3- -. . . . . . , , , . -
7
. , . , : , . , 90 . . , : , . , . , . -. , - . , . -. , , , . , . , . , - . - . . , . . - , . , , , . . . , , , . , 8
. . , . - , , . , , , . , N- , . , . . . N- , , , , , . , , . . . . . 6- , 51-56 70 90 . , , . 200. ; 6 4 , 3/2. , . - -. ( , . . ) ( ) , .
9
, . . , , , ( 99%) : , , , . ( ) . , ( ) . , , , , , , . , - , -. 90 . ( ). , . . 2*10^-5 . . 1, 6*10^-33. 0, 2*10^-13 1, 6*10^-24 . , , . - ( ), . . . 2 mi c 2 = Gmg / kompt , i mi - , - , G- , mg - , kompt - i .
10
1. (): , , . , () , , , . . , , . , , , -, , . , , . , - , . , , - . 1.1. 1.1.1. . Z = n i (arg + 2 ) n , e
-
, 0 , a , arg , K . z 1 = 0 . , , +1. +1 arg arg = 0 arg = 2i : Z 0 x , . 12
z = 2 1 e
i ( 0 + 2 )
1 2,z
=0
= 1 , z =1 = 111
1.
z = 2 1e
i ( 2 + 2 )
1 2,z
=0
= 1 , z =1 = 1
. (Z) +1. . : z ( 1)( 1) = 0 . , (-1)(-1).2
z =
( 1) e z1 = 1; z 2 = 1.
( 1) e
( i + 2 i )
1 2
(i + 2 i )
1 2
=e
( i + 2 i )
1 1 (i + 2 i ) 2e 2 ,
= 0,1
, (z) arg( 1) = i . (z) x 0 . , 1 = 0,1
2 = 0,1
z = e
(i + 2 1i )
1 1 (i + 2 2i ) 2e 2
,
z1 = ( z ) 1 =0, 2 =1 =
i i +i e2e2 i i +i e2 e2
= +1 , 1 = 2 = 0 ,
= +1 , z2 = ( z ) 1 =1, 2 =0 = 1 = 2 = 1 1 z1 = ( z ) 1 = 2 =0i 2 =e2(
= 1, = 1 .
, , +1 Z 1 . . = 0,1 , J I ,
z2 = ( z ) 1 = 2 =1 = e
i +i ) 2 2
z =
1 1 ( i + 4 k i ) + ( j + 2 k j ) 2 2, e
z1 = ( zk ) k =0 =
i j e2e2(
= ji = ji
z2 = ( zk ) k =1 = e
i j + 2i ) +j 2 e2
12
1.
= Rei + j k = ( 4i + 2j )k , k .1 = 1 , :2 2 2
,
+1, ( ji ) = ( j ) (i ) = i j = ( 1)( 1) = +1 , ji = ij .2 2
1 = ji , (1.1.)
1.1.2. . . , . . . , . .
2 + 2a + b = 0 , a, b -
2 + 2a + b + a 2 a 2 [( + a ) 2 (a 2 b)] [( + a ) a 2 b ] [( + a ) + a 2 b ] +a +a + 1 = 0 . ( a 2 b) 1 2 2 a b a b
a 2 b 0 . XY : 1) X = 0,Y 0 (1.2.) 2) X 0, Y = 0 3) X 0, Y 0 . .
1,2 = a a 2 b . . . , , .
+a2
a b
= ji2 2 2
: i = 1, j = 1, ji = ij, ( ji ) = +1 . , 13
1.
( ji 1)( ji + 1) = ( ji ) 2 ji + ji 1 = 0 . ,
3,4 = a ji a 2 b . , ( ji ) = 1 . ji . . . : 1 = 1, 2 = 3 , 2
2 + 4 + 3 = 0
2 + 4 + 3 = ( + 1)( + 3) = 0 .
3,4 = 2 ji 1 , 3 = 2 + ji, 4 = 2 ji ,
2 + 4 + 3 = ( + 2 ji )( + 2 + ji ) = 0 1 -3 ( )
2 + 4 + 3 = ( 1 + 2 ji )( 1 + 2 + ji ) = (1 ji )(1 + ji ) = 0 , 1 ji 0,1 + ji 0 . . : 1 = 0 : = 1, = 1 , 3 = ji, 4 = ji . 2
2 1 = ( + 1)( 1) = ( + ji )( ji ) = 0 . .
2 1 = (1 + ji )(1 ji ) = 0 .1.1.3. . , 14
1.
, . , . , . , . , , . , , . n
Q ( ) = cn n + cn 1 n + ... + c1 + c0 , cn , cn 1 ,.....c0 . , n
Q ( ) = ( a1 )( a 2 ) ... ( a n )
, . , , . n 2 , n 2 2 Cn =
, , . ( ai )( ak ) = ( i )( i ) , i , i - , . 3 Q ( ) = 6 + 11 + 6 = 0 , 1 = 1, 2 = 2, 3 = 3 , Q ( ) = ( 1)( 2)( 3) , 3 2
n(n 1) 2
Q ( ) = ( 2 3 + 2)( 3) = 0 ,
4 =
3 1 3 1 + ji , 5 = ji 2 2 2 215
1.
( ) , .
3 1 3 1 Q ( ) = ( ji )( + ji )( 3) = 0 2 2 2 2
Q ( ) = ( 2 4 + 3)( 2) = ( 2 ji )( 2 + ji )( 2) = 0 , , . .
, . , n 2 n >2 . . . , . . Q ( ) = ( 1)( + 1)( 2). Q ( ) = ( ji )( + ji )( 2). ,
5 1 5 1 Q ( ) = ( 2 5 + 6)( 1) = ( ji )( + ji )( 1) = 0 2 2 2 2
Q ( ) = ( 1)( + 1)( 2) = ( ji )( + ji )( 2) = 3 ji 3 + ji = ( + ji ) = ................ 2 2
,
3 ji 3 + ji Q ( ) = ( + ji ) 2 2
. n . , . .
16
1.
i i d d = Are Are (1 ji )(1 + ji ) = 0. . ,
. .
1 1 Q ( ) = ( 1)( + 1)( 2) = ( 1)( + 1)( 2) + ( ji )( + ji )( 2). 2 2
. , . .
m ( ) , Qn ( )
m n m ! n , A B + D , 2 . r ( a ) ( + p + q ) k p 2 4q 1 = , 1 , 2 , 2 + p + q 1 2 p, q 1 = , 3 , 4 . 2 + p + q 3 4 . a = 0 a. = jia a = a ( ji 1) 0 . 1 = a . a : 2 + p + q = 0 (1.2.) ( 1 )( 2 ) = ( 3 )( 4 ) = 0 , 1 , 2 ,3 , 4 , , . , 1 1 1 1 1 2 = + = 2 2 + + g 2 + + g 2 + + g 1 1 1 1 = + 2 ( 1 )( 2 ) 2 ( 3 )( 4 )
17
1.
1 2 . 3 4 . . . , , , . . , , . , . 1 1 1 1 1 1 1 1 1 = + 2 ( ) 2 + p + q 2 (1 2 ) 1 2 3 4 3 4
, k , , , . r 0 r . (1 ji ) . 1.1.4. = z + j (1.3.) z - +i, +i, i, j - , :
= k + re i + j (1 ji ) -
= re i + j (1 ji )
, , (1.4.) =z+j=(+i )+j(+i).
ii=jj=-1 ij=ji=k, (ij) 2 =(ij) 2 =k 2 =1
z
z=Re =Re (z+j ) =Im =Im (z+j )
, , , :
=Re Re , =Im Re ,
=Re Im =Im Re
18
1.
= 0, z. z = 0, , = j. , : z1 + j 1 = z2 + j 2 , z1=z2, 1=2. 1=-2, 2
z + j = z j =
. 1. . 1+2 1=z1+j1 2=z2+j2 =1+2=(z1+z2)+j(1+2). 1-2. =1-2=(z1-z2) + j(1-2) :
2. . 12 -
1+2=2+1; 1 +( 2 + 3 )=( 1 + 2 )+ 3 1 =z 1 +j 1 , 2 =z 2 +j 2
, , , z . , : 12=21; 1(23)=(12)3; (1+2)3=13+23. 1.1.5. (), . . , . , . , , . , , .
= 1 2 =(z 1 z 2 - 1 2 )+j(z 1 2 + 1 z 2 ). 1=j, 2=j, jj=-1.
19
1.
- . , . , , . +
. 1. . , . : , , , , . , . , , , . 0 + 0 = . , , , , 0 , . () . , . , . . . . , 0 = 0ei , , , . , .iy
x
. 2. , . , . 0 = 0ei , . . , , ,
20
1.
. . , . C3 . , : R- , r . C3 4 , 2 . C3 . , , . +1 4 I 2 J. (. 1.1.1.) , , : , , .
. 3. . [1], [2], [3], [4]. , , . z , (1.3. 1.4.).
z = e i ,
= re i , = z,r=; , - . z, , : , r - z, :
21
1.
z = = x2 + y2 ;
= r = 2 +2 , = arg z, = arg , , 2:
y + 2 k ; x = arg = arctg + 2k .
= arg z = arctg
, = z + j = ei + jrei . i j ( ) ij=ji, :
= x + iy + j + ji = ( x + i ) + j ( y + i ) = 1e j1 + ir1e j 1 , 1 = x + j ;r1 = y + j ; :
1 = arg( x + j ); 1 = arg( y + j ). , . , ,
= e i + jre i = R1e j . R1 ,
R1 = z 2 + 2 ;
- ,
= arg = arctg
r i ( ) e ,
, . .
= R1 = R = 4 4 + r 4 + 2 2 r 2 cos 2( ) , - 2 sin 2 + r 2 sin 2 1 = arg R1 = arctg 2 . , 2 cos 2 + r 2 cos 2
= Re ia + j .
(1.5.)
22
1.
R, - ; - . .
. 4. . .
=Re
i+j
iy
x. 5. . Y.
23
1.
(. 4) (1.3.) i i z=e j=jre , r (z). i j=jre -
r +
0 arg 2
, , >0, , (z). , j , . i j : e , (z), jre , . . z, .
a) ) . 6. : - ; . , ,
= 2 + r 2 ei + ja , r a = arctg - . . 5, a . r, , ;
= ei + jrei = ei ( + jr ).
24
1.
, jre , , , (z). : i i z=e , =jre
i
. 7. : .
/2
. 8. : . = R a,
/ 2 a / 2, 0 2 . j (. 6, . 7..). () R, , a. 25
1.
(1.5.) :
x = R cos cos a; y = iR sin cos a; z = j Rei sin a.
.( , , ). 1.1.6. . . a v (1.5.),
r a = arctg ( ei ( ) ) = S + i ,
= Rei + j = Rei + jS + ji .
(, a - )
(1.6.)
= ( x + iy ) + j ( + i ),
(1.6.) v R, , S, - . (1.6.).
R = = 4 x 4 + y 4 + 4 + 4 + 2 x 2 y 2 + 2 x 2 2 + + 2 y 2 2 + 2 2 2 2 x 2 2 2 y 2 2 + 8 xy, , = = 0 , S==0,
R = = x2 + y 2 ,
= ei ., R (1, i) i, , ; y==0, =x+j, = 0, = 0, R = Rz = x 2 + 2 , = Re jS , , S R2 R (1, j) 1; y==0, i jarctg 2 2
= x + ji = x eR = R3 = ,
;
(6) =0, S=0,
= R3e ji .
, - R3 R (1, ij) 1. (1.6.) , . ,
26
1.
ei + jrei ei + jrei 4 4
(1.7.)
4 + r + 2 2 r 2 cos 2( ) 2 + r 2 . , , , Z, , , =.
C0 C1
K g 2
C2
1
. 9. - () - , (z). > 0, , (z), . 4, 6. arg z = arg , , , 0. (v) . ,
= z , arg z arg = . 2
(1.8.)
27
1.
. 6, . 8 . . 7 . (1.8.)
= ei jirei = ei (1 ji ) = jei ( j j ) = jei ( j i ) = " i je () , -
= e
, , (z) /2. r= (. 8). - (. . 9, 10, 11). i - e (i j ) , , r=, - /2. , . 9, 10, 11. . y1, 1, 2 (1.8.). - , : i i
= ji ( + )e ,
1
1
= e
. 10. i + i i
+ je .
28
1.
- 0 , . arc tg i, (1.5.), (1.8.). (1.9.) = e i (i j ) = 0 ei e jarctg (i ).
(9) . ij=ji (1.10.) (i + j )(i j ) = ii + ji ji = 1 + ji ij + 1 = 0. . , .
i + j 0, i j 0,
(i + j )(i j ) = 0.C1
C2
C1 C2
. 11. - (1.9.). (1.9.) :
29
1.
= ei (i + j ) = 0 ei e jarctg (i ) 0;
= ei (i j ) = 0 ei e + jarctg (i ) 0. ,
arg = arctg (i ). . , arctg i, . , . (1.10.) i, j , .[5] , (1.11.) jarctg (i ) + jarctg (i ) 0 , . , . , , . . , . - , z, (). z z=, arg z arg = /2 , /2. .1
= 0;
(i + j )(i j ) =
0e
0e
= 0 0e = 0.
2
. 12. , . 8 . , , -
30
1.
/2. , , . ( ). , , , , . arctg i. . (. 12, 2 ). 1 , . 1.1.7. () , , - . . , (1.12.) a 0 = b0 = 0, a b . (1.12.) , , , , (1.12.) . , , . . . , . () , (1.12.) . () : (1.13.)
= ( x + iy + j + ji )(i + j ) = [( x y ) + i ( y )](i + j ); = ( x + iy + j + ji )(i j ) = [( x + ) i ( y )](i j ).
(1.13.) . (1.13.) (1.12.). =ij , . , ,
1 = e jarctg (i ). , () , (1.11.), .
a+0=a
+ = + i j
31
1.
1 ~ , 0
1 = e # jarctg (i ). i j 0 ~ 0e , i + jS + ji
,
= a + ib + jc + jid ; ' = a '+ib'+ jc'+ jid ' ; (1.14.) = x + iy + j + ji. , (1.14.), :
, S, . , . :
= ' (1.14)
ax by a + d = a; bx + ay d c = b' ; cx dy + a b = c' ; dx + cy + b + a = d '.
(1.15.)
(1.15.) v':
Re Re( ) = Re Re ' = a ' ; Re Im( ) = Re Im ' = b' ; Im Re( ) = Im Re ' = c' ; Im Im( ) = Im Im ' = d '.
, . (1.15.) , . ,
a b c d
b
a d c
c d a b
d c 4 = . b a
, (1.14.) , (1.15.) . () (1.15.) : = 0, a=0, b=0, c=0, d=0; 32
1.
= 0 , . , , , (1.13.). , , , , , . () , [6] , (z) . () . (1.13.). , , () , (z) . :
1 2 = 1 2 .
[5], , , . , - . 1, i, j, k :
i 2 = j 2 = k 2 = 1; ij = ji = k ; jk = kj = i; ki = ik = j. . , , (z) . . , . , , .
d = Rei + j ei1 (1 ji ) = Rei + j 0e i1 jarctgi == R 0e i +i1 + j jarctgi arctgz + arctgz1 = arctg
z z1 , 1 z1 z
33
1.
r i ( ) i e artgi = arctg = r i ( ) i 1 e r i ( ) i e = arctg = arctgi , r i ( ) i (i e )
(1.16.)
, .
d = Rei ( +1 ) (1 ji )
(i (i (i (i (i n
j ) 0 = 1 , , j )1 = i j j ) 2 = 2i (i j ) j ) 3 = 4(i j ) j ) 4 = 8i (i j )(1.17.)2
(i j ) = (2i ) n 1 (i j )
(i j ) = 2i (i j ) . , , .
(i j ) 2 = 2i (i j ) (i j )(i j 2i ) = 0 . . i j 0, i j 2i . , , , . ,
(i j )5 = 8i(i j ) 2 (i j ) 4 = 8i (i j ) .
(i j ) 2 = 2i (i j ) (i j ) = 2i (i j ) (i j ) =
(i j ) . 2i
. , . , , .
34
1.
0 = 0e arctgi .
i
, 0 = 0e
i + j
,
, . 1.1.8. , . i, j ij=ji : (1.19.) 1 = ( x + iy ) + j ( + i );
d = Rei + j (i j ) , arctgi .
(1.18.)
2 = ( x + j ) + i ( y + j ).
(1.20.)
(1.19.) , (z), (1.20.) (i, j). (. 13.), .e j arc tg i
j2e
j3
1
j2ei2 i jre i
jr1e ji 1e ji i eii i 1e 1 i 1 2
e j arc tg j
1ei1
. 13. . e e iactg ( j ) jarctg (i )
, - -
.
35
1.
(1.6.) (1.19.), (1.20.) :
1 = R1ei1 + jS1 + ji 1 ; 2 = R2 ei2 + jS2 + ji 2 ;, . (1.19.), (1.20.) . , R1=R2, 1=2, S1=S2, 1=2. . (1.19.), (1.20.):
1 = ( x + iy ) + j ( + i ) = ( x 2 y 2 + 2 2 ) + 2i ( xy + ) ;R1 = 1 = 4 x 4 + y 4 + 4 + 4 + 2 x 2 y 2 + 2 x 2 2 2 x 2 2 2 y 2 2 + 2 y 2 2 + 2 2 2 + 8 xy; 2 = ( x + j ) + i ( y + j ) ; R2 = 2 .
1 = 2 ; R1 = R2 .
, :
1 -
2 xy + 2 1 = arg 1 = 1 arctg 2 2 x y 2 + 2 2 .
S1 1, , 1 jarctg
1 = 1 e
+i x +iy
. 1+ i
+ i + i 1 x + iy = ln arctg . x + iy 2i 1 i + i x + iy
S1, 1. :
S1 =
1 2 x + 2 y jarctg 2 ; 2 x + y2 2 22
ln 1 1 1 = ji . 2 (x + )2 + ( y )2
36
1.
(1.20.), . , S2 2
1 2 x + 2 y S2 = arctg 2 . 2 x 2 + y 2 2
, S1, S2=S1 . , . . . =ij=ji , = R = 1 S =/2, =/2, =0.
37
1.2. : , , , , (z). , [7]. , , ( ) . (Y) , (z), - . , (), . 1.2.1. (Y) , G (Y) (Y). =f(),
= x + iy + j + ji , = U + iV + jP + jiR.
, :
U = U ( x, y , , ) = Re Re (v ); V = V ( x, y , , ) = Re Im ( v ); P = P ( x, y , , ) = Im Re ( v ); R = R ( x, y , , ) = Im Im (v ).
, [7]. (z). ,
v = e i + jrei = z + j ,
f ( v ) = W ( z, ) + jT ( z, ),
:
W ( z, ) = Re f (v ); T ( z, ) = Im(v ).
0 ,
f()
38
1.
z 0 0
lim W ( z, ) = W0 ; lim T ( z, ) = T0 ;
z 0 0 , ,
v v0 (z) v v 0 . , v v 0 f() f(0). 0, 0. , - . f(), , , (1.21.) f (v + h ) f (v)
lim f ( v ) = W0 + jT0 = f ( v 0 ).
h h0 , (). f() W T : 1. f()=W(z, )+jT(z, ) , , W (z) W W T T , , , , z z
lim
= f ' (v ).
, :
- . (1.22.).
W T = ; z T W = . z f ' ( v ) = lim
(1.22.)
h . . +h z=x+iy.
f (v + h) f (v) . h h 0
39
1.
. , +h j, t0 h=jt, t = + i .
W ( z + S , ) W ( z , ) + S S 0 T T ( z + S , ) T ( z, ) W + j lim = +j . z z S S 0 f ' ( v ) = lim
W ( z, + t ) W ( z, + t ) it t 0 W T T ( z, + t ) T ( z, ) + j lim =j + . jt t 0 f ' ( v ) = lim f ' (v ) = W T W T + j =j + . z z
,
. (1.22.). x, y, ,. , ). . : i i ) v = e + jre ; i i ) v = e + jre . ). f() i i i i
f ( v ) = W ( e , re ) + jT ( e , re ).
, +h i i i i
h = dv = de
+ ie d + jre id + jdre
=
= dei + jdrei + iei ( + jr )d . d 0, dr = d = 0; dr 0, dr = d = 0; d 0, dr = d = 0. (1.21.) h0:
j; - i; - , . =const, r = const,
40
1.
W ( + )ei , re i W e i , rei + f ' ( v ) = lim 0 e i T ( + )e i , re i T ei , re i + j lim = 0 e i W T = e i + je i . W [e i , ( r + r )e i ] W [(e i , re i )] + f ' ( v ) = lim r 0 jre i , =const, =const,
[
] [(
)]
[
] [(
)]
(1.23.)
T [e i , ( r + r )e i ] T [(e i , re i )] + j lim = z 0 jre i W T = je i + e i . r r
(1.24.)
, = const, r = const, W ei + iei , rei + irei W (ei , rei ) + f ' ( v ) = lim 0 i ( ei + jrei )
[
]
+ lim j 0
T ei + iei , rei + T ei , rei + ei
[
] [
]
T e i W + +j = i( + jr ) W T W T +r r . . = ie i + jie i . 2 2 +r 2 + r2
(1.25.)
(1.23.), (1.24.), (1.25.) f()
f ' ( v ) = e i
= ie
i
W T W T + je i = je i + e i = r r W T W T +r . . i + jie 2 + r2 2 + r2
(1.26.) ,
:
W T W + rT = = i ; r 2 + r 2 T W rW T = =i . r 2 + r 2
(1.27.)
41
1.
f , :
e i e i
W T W 1 i T i = e i = ie i = e ; r r T W i T i i i W = e i = . e = e r r r
(1.28.)
f ' ( v ) = e i = 1 i ie
W T T W + je i = e i je i = r r W i T i j e =!
(1.29.)
(1.28.), (1.29.) . (1.22.), (1.27.), (1.28.) , . (z) ( ). . , , - . +h i
h = e (i j ),
0 . 0 , (i j ) . e (i j ) .i
f v + ei (i j ) f (v ) lim . 0 ei (i + j )
[
]
(1.30.)
(z) z=0. h=0 i e (i j ). (Y) . 0 = 0 + z (1 ji ) . , 0 , . 0 = z (1 ji ) , 0 . z , lim z 0 ( 0 ) = 0 , lim z ( 0 ) = (1 ji ) .
42
1.
f ( ) , = (1 ji ) . f = f [ 0 + (1 ji )] f ( 0 ) ,
f / ( 0 ) = lim 0 lim 0
f [ 0 + (1 ji )] f ( 0 ) , (1 ji )
f = f / ( 0 ) . (1 ji )
f = f [ 0 + (1 ji )] f ( 0 ) =
f 1 2 f 2 (1 ji ) + (1 ji )2 + ..... + 0( (1 ji )) 2 0 2 0 , f (1 + ji ) 0 = f ( 0 ) lim 0 (1 + ji ) , " 0
= ( ) " 0 ,
f f / ( 0 ) # , (1 ji )
0 # (1 ji ) #
0( (1 ji )) , . ,
f = f / ( 0 ) (1 ji ) + 0( (1 ji )) 0 , A f ( )
f = A (1 ji ) + 0( (1 ji )) , ./
0 A = f ( 0 ) 1.2.2. . , . - . , . . , .
43
1.
. : .) , ; ) ; ) ; ) . n A. = v = n v n - , (Y). n () - : iF + j n in + jn
Re
=r e
,
(1.31.)
R = r n , F = n , = n , . (1.6), iF + j + jiS n in + jn + jin : n
Re
=r e
R = r , F = n , = n , S = n ,
(1.32.)
. (1.31.), (1.32.) , , n, , , (n-1) arg n 1 v = r r .
= 2 n = 2 n
= 2 n
. 14.
Re iF + j = r n e in + jn . R = r n , F = n , = n .
44
1.
(1.31.) , - . r ,
0 0 , f ( ) f ( ) < (1 ji ) , < (1 ji ) , = ( ) >
0,
> 0 . f ( ) f ( ) 1 1 J1 d < d . 2 3 2 3 d = 2 3 ,
d = 2 3 (1 ji ) ,
79
1.
< (1 ji ) .
0 0 , , . . G S , S1 , , 1 , 1 f ( ) 1 f ( ) d 4i + 2j d 4i + 2j 1 , 1 . f ( ) = 1 f ( ) f ( ), G d = 0, G . 4i + 2j 1.3.5. 1 f() G (), , , f (v )dv . C . z. 1 3 , , G (), G .
J 1 < .
S, .
v = e i + jrei .
i i i
dv = e d + je dr + ie d + jirei d . f (v ) = W e , re
f() i i i i l =
(
)+ jR(e
C
f (v )dv .
, re
).
l = l1 + jl2
l1 = Wei d iWei d R ei dr iRrei d ;C
(1.44.)
80
1.
l2 = Wei dr + iei Wrd R ei d iRei d .C
(1.45.)
l1 l2 . , , . , . . , :
l1 = We i iWe i
dd +
+ We i R e i ddr + r i i + We iR e r dd + + iWe i R e i drd + r + iWe i R e i r dd +
+ R e i + iR e i r drd ; r i l 2 = We r Wr drd + + We i R e i ddr + r + We i ie i R ddr + r
(1.46.)
i + ie i rW e R dd + + Wie i r ie i R dd + i i + R e r ie R dd .
(1.47.)
(1.44.), (1.45.) (1.28.), (1.29.). ,
81
1.
(), . . i i
v = e
+ jre ,
Ci i . . C0, C1, C2 (. 19) , z,
f (v ) = W ( ei , re i ) + jR( ei , re i ). l = f (v )dv
r = const; = var; = var .
i i
dv = ie d + e d + jirei d .
l=
C0
(W +
jR )(ie i d + ei d + jirei d ) = l1 + jl2 =
= Wiei d + Wei d Rirei d + + j Riei d + R ei d + iWrei d .C0
C0 l1, l2: l1 = Wiei d + Wei d Rirei d = C0 Wei i i
We i Rire dd = W i R i W i i i = e + ie W ie We i + ire = W i W i R i = e ie + ire C0
(
)
dd =
l2 =
(Rie
i
iWrei d + R e i d = dd =
)
dd ;
R R W i = e i + ie i i re
R R W i = e i + ie i R ie i R e i i i re dd .
82
1.
.
P1 =
W W R i + ir; R R W i + ir P2 = ;
(1.48.) (1.49.)
, 0 , . (1.46.), (1.47.) , (1.27.), . . C5, = const (. 20). i i i dv = ie d + jire d + jdre . .
l=
C5
f (v )dv = (W +C5
jR )( iei d + jirei d + jdrei ) = l1 + jl2 =
= Wiei d + jrei Rd ei Rdr + + j iei Rd + irWei d + Wei Rdr.C5 C5
=const
C5
. 25. C5 , . :
l1 = ( iei W irei R )d ei Rdr =C5
83
1.
R W R = e i ie i + ire i ddr = P3 ddr; r r i i i l2 = (ie R + ire W )d + e Wdr =C5
W R R = e i ie i R ie i + ire i + ie i R ddr = r r
P3 4 :
W R W = e i ie i ire i ddr = P4 ddr. r r W R R i P3 = i r + ir r e ; W R W ire i ie i P4 = e i . r r
W W R = e i + ie i W ie i ire i ie i W drd = r r
(1.50.) (1.51.)
, (1.27.). . . : C = const . i i
dv = de
+ jdre ;
l=
C
(W +
jR ) de i + jdre i = l1 + jl 2 =
(
)
(1.28.), . . . C3. i i i i
R R W W = e i e i e i ddr + j e i drd . r r
dv = de
+ ie d + je dr + ije rd .
l=
= (Wei d + iei Wd ei Rdr iei rRd ) +C
C3
(W +
jR ) de i + ie i d + je i dr + jie i rd = l1 + jl 2 =
(
)
84
1.
C l1 l2.
+ j e i Rd + ie i Rd + e i Wdr + ie i rWd .
l1 = ei Wd e i Rdr (ie i W irei R )d =C3
R W = e i e i ddr + r
W R R + ie i ire i ie i R + ie i R + e i ddr + r r W W R + ire i + ie i W + e i ie i W ie i dd ; C3
l2 =
e
i
Rd + ie i Rd + e i Wdr + ie i rWd =
R W W + ie i + ie i W + ire i ie i W e i drd + r r R R W + e i + ie i R ie i R ie i ire i dd . 1
W R = e i e i ddr + r
2
2
C3
. 26. P2, P1, (1.28) (1.28.). P4, P3, (1.28.) (1.23.).
85
1.
: 3 r (. 26). , l1, l2 . C3. . , -.
R W e i l1 = e i r V +
i W i R i R + ie r ire r + e +
i W W R ie i + ire i = e r 2R R 2W W = ei iei iei iei + r r V 2 2 2 i W i W i R i R + ie + ie ire +e + r r r 2 2 2 i W i W i R i R +e ie + ie + ire dddr = r r r = 0dddr = 0;V
W R i R W W e i + ire i e i l 2 = e i + + ie r r r V R R W + e i ie i ire i dddr = r 2 2 i W i W i R i R = ie +e ie e + r rd V
V , .
+ 2R 2R W 2W + ei i iei irei dddr = r r r = 0 dv = 0; r r r
+ ie
i R
+ ie
2 i R
+ ire
2 i W
e
2 i W
86
1.
() - a , : C3 V *
a dr = rota d
= divrotadv = 0
. : 2. 3 - ( ) () , , , , , . f (v )dv . C3 3. f() G () 3 G . 0, , , 3
divrot a = 0.
f (v 0 ) =
1 f ( )d v 4i + 2j C 0 3
(1.52.)
. 3 f ( )d = f ( )dC3
r 3 (. 26) f() () G , 0. - , 0. ,
v0 = e i + j (i j )
. . -0=0. r , 3, . 25. v 0 = R , :
J=
1 f ( )d 1 f ( )d = = 4i + 2j r v0 4i + 2j C R v0 f ( ) f (v0 ) + f (v0 ) 1 d = v0 4i + 2j CR
=
f (v0 ) f ( ) f (v0 ) 1 d v d + 4i + 2j v = J 1 + f (v0 )J 2 . 4i + 2j C R CR 0 087
1.
J2 J1:
1 d 1 R e i + j (id + jd ) = = J2 = 4i + 2j C R v0 4i + 2j C R R e i + j
f() 0 >0 = ( ) ,
2 4 1 = i d + j d = 1. 4i + 2j 0 0
f ( ) f (v0 ) < .
, v0 < . , R < ,
l1
