komplekna vizija
DESCRIPTION
o nekim dobrim vizijamaTRANSCRIPT
-
..
.
2010
-
517.5
-
7 16 2010.
8 14 2010.
.-. , .. ..
:. .-. , ..
. .-. , () ..
.. ( ):
/ .. . : , 2010. 102 .
, - . - .
c , 2010c .., 2010
-
-
- -. () ( ). , , .
3
-
1. .
z = (x, y), x, y R, .. R
. y = 0, (x, 0) x, - x.
1.1
1. : z1 +z2 = (x1 +x2, y1 +y2), z1 = (x1, y1), z2 = (x2, y2).2. : z1z2 = (x1x2 y1y2, x1y2 + y1x2).. (0, 1), -
i - .
i2 = i i = (0, 1)(0, 1) = (1, 0) = 1 i = 1. i, -
:
z = (x, y) = (x, 0) + (0, y) = (x, 0) + (y, 0)(0, 1) = x+ yi = x+ iy
( ), x - z (x = Re z), y - z (y = Im z).
3. : z1 z2 = (x1 x2, y1 y2). 1. z1 = z2, x1 = x2 y1 = y2. 2. z = (x,y) = x iy
z = (x, y) = x+ iy., z z = x2 + y2.4. z1 z2
z , z1 z = z2. z1 z = z2 z1
x21 + y21
, z =z2 z1x21 + y
21
=z2z1,
z1 = x1 + iy1 6= 0.
4
-
1.2
- :
1. :
z1 + z2 = z2 + z1, z1z2 = z2z1.
2. :
(z1 + z2) + z3 = z1 + (z2 + z3), (z1z2)z3 = z1(z2z3).
3. :
z1(z2 + z3) = z1z2 + z1z3.
. z1=x1 + iy1,z2=x2 + iy2. z1 + z2=(x1 +x2) + i(y1 + y2), z2 + z1=(x2 +x1) + i(y2 + y1). x1 + x2 == x2 + x1, y1 + y2 = y2 + y1. , z1 + z2 = z2 + z1.
1-3.
1.3
6
y
-xo
7rz = (x, y)
z = x + iy - (x, y), z. - , - . , - . , , - (: C).
z o z.
5
-
1.4
=x2 + y2 =
z z - z = x + iy.
6
y
-xo
7rM(x, y)
, - OM , z. -: = |z|. , OM x, z = Arg z. , , - 2pi:
Arg z = arg z + 2kpi, k = 0,1,2, . . . , arg z - Arg z, pi arg z pi,
arg z =
arctgy
x, x > 0,
pi + arctgy
x, x < 0, y 0,
pi + arctg yx
, x < 0, y < 0,pi
2, x = 0, y > 0,
pi2
, x = 0, y < 0.
. 1. , . - .
2. z = (x, y), :x = cos, y = sin. : z = (cos+ i sin).
1.5
. |z1 z2| = |z1| |z2|, Arg (z1 z2) = Arg z1 + Arg z2;z2z1 = |z2||z1| ,
Argz2z1
= Arg z2 Arg z1.. z1 = (cos + i sin), z2 = r(cos + i sin).
z1z2 = r(cos+i sin)(cos+i sin) = r(cos(+)+i sin(+)). .
( ) - . , z2 =
z2z1z1, z1 6= 0. |z2| =
6
-
z2z1 |z1| z2z1
= |z2||z1| . Arg z2 = Arg z2z1 +Arg z1 Arg z2z1 = Arg z2Arg z1. .
1. |zn| = |z|n,Arg zn =nArg z, :
(r(cos+ i sin))n = rn(cosn+ i sinn).
2. - , - : , - : z1 = |z1|(cos1 + i sin1), z2 =|z2|(cos2 + i sin2), z1 = z2 , |z1| = |z2| 1 = 2 + 2kpi, k Z.
1.6
1. |z| = |z|; 2. z z = |z|2;3. |z1z2| = |z1| |z2|; 4. |zn| = |z|n;
6
-
HHHHHHHHj
HHHH
HHHHY
1
BBBBBBBBBBBBM
rz2
rz1 z2rz2
rz1rz1 + z2
HH
HH
HH
HH
HH
o
.
5.z1z2 = |z1||z2| , z2 6= 0;
6. |Re z| |z | , | Im z| |z |;7. |z1 + z2| |z1|+ |z2|;8.|z1| |z2| |z1 z2|.
1 2 z -. 3, 4, 5 - - -. 6, 7,8 - (. ).
1.7
. z = (cos + i sin), = |z|, = arg z, nz , ,
n, z: ( nz)n = z. n
z, , n,
+ 2kpi
n, k = 0,1,2, . . . . k
k = 0, 1, . . . , n 1, n nz.
7
-
,
nz = n(cos
+ 2kpi
n+ i sin
+ 2kpi
n), k = 0, 1, . . . , n 1.
n nz -
n-, - n.
1.
1) (2 + i)3 2)3 + i
(1 + i)(1 2i)
. 1.) (2 + i)3 = 8 + 3 22 i+ 3 2 i2 + i3 = 2 + 11i.2.)
3 + i
(1+i)(12i) =(3 + i)(1 + i)(1 2i)
((1 + i)(1 2i))((1 + i)(1 2i)) =(3 + i)(3 i)|3 i|2 =
=(3 + i)2
10=
9 + 6i 110
=4
5+
3
5i.
2. , z1 + z2 = z1 + z2.. z1 + z2 = (x1 + x2) + i(y1 + y2) = x1 + x2 i(y1 + y2) =
= (x1 iy1) + (x2 iy2) = z1 + z2. 3.
(4 + 2i)x+ (5 3i)y = 13 + i.
. - : (4x+ 5y) + i(2x 3y) = 13 + i. {
4x+ 5y = 13,
2x 3y = 1. , x = 2, y = 1.
4.
z = sin pi10 i cos pi
10.
. x = sin pi10
< 0, y = cos pi8< 0. arg z = pi +
+ arctg
cos pi10sin
pi
10
= pi + arctg (tg (pi2 pi
10
))= pi + arctg
(tg
2
5pi
)=
= pi + 25pi = 3
5pi, Arg z = 3
5pi + 2kpi, k = 0,1,2, . . . .
8
-
|z| =
sin2pi
10+ cos2
pi
10= 1.
5. z = 1 i3.
. |z| =
(1)2 + (3)2 = 2; tg =
3
1 =
3, = pi ++ arctg
3 = pi + pi3 = 23pi. 1 i
3 = 2(cos (23pi) + i sin (23pi)).
6. (1 + i3)60.. 1 + i3
1 + i
3 = 2(cos 23pi + i sin23pi).
(1+i3)60 = 260(cos (6023pi)++i sin (6023pi)) = 2
60(cos 40pi + i sin 40pi) = 260. 7. 4
1 i.
. 1 i
1 i =
2(cos (pi4) + i sin (pi
4)).
:
4
1 i = 8
2(cos ( pi4 + 2kpi
n) + i sin (
pi4 + 2kpin
)).
k = 0, 1, 2, 3, 4
1 i = 8
2(cospi
16 i sin pi
16) , k = 0;
4
1 i = 8
2(cos7
16pi + i sin
7
16pi) , k = 1;
4
1 i = 8
2(cos15
16pi i sin 15
16pi) , k = 2;
4
1 i = 8
2(cos23
16pi i sin 23
16pi) , k = 3.
8. -
: 1 |z + i| 2, pi4< arg z 0, N = N() N, n > N : |zn z0| < .
. (zn), , .
1. lim zn = z0 , xn x0, yn y0( zn = xn + iyn, z0 = x0 + iy0).
. . znz0. n>N : |zn z0| N. xn x0, yn y0.. xn x0, yn y0 n > N :
|xn x0| < 2, |yn y0| <
2. |zn z0| =
(xn x0)2 + (yn y0)2 < ,
n > N. zn z0. . 2. (zn) ,
M > 0 , |zn| M , n N. 2. (zn) . 3. lim zn = a, lim zn = b (a, b C), 1. lim (zn zn) = a b2. lim (zn zn) = ab
16
-
3. limznzn
=a
b(b 6= 0).
- , .
3. E > 0, N = N(E) N, n > N : |zn| > E, , (zn) , lim zn =.
C - z =.
4. C{} -
. 5. -
{z : |z| > E}. z G,
G.
G ,
1) G .2) ,
. 6. z0 -
, ; - z0 - , - |z z0| < .
7. W f(z) z0 (: lim
zz0f(z) = W ),
1) f(z) z0, z0.
2) (zn) :
zn z0, zn 6= z0, f(zn) W.. 2) 7 -
:2) > 0, > 0 : |z z0| < , z 6= z0, |f(z)W | < . 4. f(z) = U(x, y) + iV (x, y), z = x+ iy, z0 = x0 + iy0.
limzz0
f(z) = W = a+ ib , lim(x,y)(x0,y0)
U(x, y) = a,
lim(x,y)(x0,y0)
V (x, y) = b.
. zn z0, zn 6= z0 (zn = xn + iyn).limzz0
f(z) = W = a+ ib, f(zn) a+ ib.
17
-
f(zn) = U(xn, yn) + iV (xn, yn) a + ib. 1 - , U(xn, yn) a, V (xn, yn) b. .
8. f(z) z0, f(z) z0 lim
znz0f(z) = f(z0).
5. f(z) = f(x+ iy) = U(x, y) + iV (x, y) z0 = x0 + iy0 , U(x, y), V (x, y) (x0, y0).
4. 9. f(z) G
C, . 6. f(z) g(z) G.
f(z) g(z), f(z) g(z), f(z)/g(z), g(z) 6= 0, G.
1. f(z) = C, C - const C. C.
2. f(z) = z, z C - C.3. f(z) = az + b (a, b C) - C.4. f(z) = zn, n N - C.5. f(z) = ez, z C.ez = ex+iy = ex(cos y + i sin y) = ex cos y + iex sin y. U(x, y) =
= ex cos y, V (x, y) = ex sin y. U(x, y), V (x, y) R2,, ez C (. 5).
6. cos z =eiz + eiz
2, sin z =
eiz eiz2i
, z C.cos z =
1
2(eixy + eix+y) =
1
2(ey(cosx+ i sinx) + ey(cosx i sinx)) =
=1
2(ey cosx+ ey cosx) + i
1
2(ey sinx ey sinx).
U(x, y)= cosxey+ey
2= cosx ch y, V (x, y)= sinx e
yey2
= sinx sh y.
U(x, y), V (x, y) R2, , cos z - C (. 4). sin z C.
7. ( 6) - C
ch z =ez + ez
2, sh z =
ez ez2
.
18
-
1. lim zn, -
zn =n in+ i
, n N, z0 = 1.
. > 0. , - N N , n > N , |zn 1| < .
|zn 1| =n in+ i 1
= 1 + in+ i = 2n+ 1 < . n >
2
n 1.
N N = N() = [
2
n 1]. [x]
x. 2. , zn z0, |zn| |z0|.. zn z0,
lim |zn z0| = 0. (1) zn z0
||zn| |z0|| |zn z0|. (2) (1) (2) , |zn| |z0|.
zn = nein, n = |zn|, n = arg zn. , lim n = 0,limn = 0, lim zn = 0ei0.
3. ,
limn
(1 +
z
n
)n= ez, z = x+ iy.
. zn =(
1 +z
n
)n. lim |zn| = lim
n
(1 + zn
)n== lim
n
((1 +
x
n
)2+y2
n2
)n2
= limn
(1 +
x2 + y2 + 2xn
n2
)n2
= ex.
arg zn = arg (1 +z
n)n = n arg (1 +
z
n) = n arctg
yn
1 + xn= n arctg y
n+ x.
limn arg zn = limnn arctg
y
n+ x= y.
,
limn
(1 +
z
n
)n= ex eiy = ex+iy = ez.
19
-
:
78. zn = (1 +1
n)ei
pin . 79. zn =
in
n. 80. zn = (1 + 3i)
n.
81. zn =ein
n2. 82. zn =
n+ 2i
3n+ 7i. 83. zn = e
i(pi2 + 12n ).
84. zn = n sini
n. 85. zn=n cos
npi
2+ in sin
npi
2. 86. zn =
sh in
n.
, , :
87. limz1
2z + 1
z + 2= 1. 88. lim
z34i|z| = 5.
:
89. limzi
z2 + 3iz 2z + i
. 90. limzpi4
cos 2z
ch iz + i sh iz.
91. limz0
sin z
sh iz. 92. lim
z pi2 i
e2z + 1
ez + i.
, :
93. f(z) = z. 94. f(z) = Re z. 95. f(z) = Im z.96. Pn(z) = a0zn + a1zn1 + + an, ak (k = 0, 1, 2, . . . , n) - -
.97. , R(z) =
P (z)
Q(z), P (z), Q(z) - , -
C, Q(z) 6= 0.
4. . - w = f(z)
G C. z z+ z G. w =f = f(z + z) f(z), z = x+ iy.
1. w = f(z)
z G, wz
, z 0 . w = f(z) z f (z),
f (z) = limz0
w
z.
20
-
1. w = A z + o (z), z 0, A - - , z, - f(z) z, A = f (z).
, - .
2. w = f(z) , : dw = f (z)dz, dz = z - z.
2. w = f(z), z G, G, z G.
. c f(z) z G - f(z) z, G.
1. f(z) = C (C - const) C (C) = 0.
.
limz0
f
z= lim
z0C C
z= 0, .. (C) = 0.
2. f(z) = zn (n 1 - ) C (zn) = nzn1.
.
limz0
(z+z)nznz
= limz0
zn+C1nzn1z+C2nz
n2(z)2+ +Cnn(z)nznz
=
= limz0
nzn1z + o (z)z
= nzn1.
3. w = zRe z z = 0 z 6= 0.
. z = 0:w
h=hReh
h= Reh = h1 0,
h = h1 + h2i 0. , w = zRe z z = 0. z 6= 0. w
h=
w(z+h) w(z)h
=(z+h) Re (z+h) zRe z
h= z
Re (z+h) Re zh
+
+ Re (z + h). h = h1 0, Re (z + h) Re zh
=x+ h1 x
h1 1
w
h2x + iy h = h10. h = h2i0. Re (z+h)Re z
h=
=x xh2i
= 0 h2.
21
-
,w
h x. ,
h w
h . -
, w = zRe z z 6= 0. 4. w = Re z C.
.w
h=
Re (z + h) Re zh
h = h1 + h2i 0 z 6= 0 (. 3). z = 0. w
h=
Reh
h=h1h. h = h1, h2 = 0,
w
h 1 h = h1 0.
h = h1, h2 = 0, wh 1 h = h1 0. ,
w = Re z z = 0, 4.. 1.) w = zRe z w = Re z
z C.2.) w = zRe z z = 0.
( -). f(z) = U(x, y) + iV (x, y) z = x + iy, U(x, y), V (x, y) (x, y) R2, z = x+ iy. f(z) z ,
U
x=V
y,U
y= V
x, ()
-.. .
f (z) = limh0
f(z + h) f(z)h
. , f (z) h 0. h = s 0, s R, .. 0 .
f (z) = lims0
U(x+ s, y) U(x, y)s
+ i lims0
V (x+ s, y) V (x, y)s
=
=U
x+ i
V
x.
(1)
, f (z), , .. h = it 0, t R.
f (z) = limt0
U(x, y + t) U(x, y)it
+ i limt0
V (x, y + t) V (x, y)it
=
= iUy
+V
y.
(2)
22
-
(1) (2), ().
. U(x, y) V (x, y) (h = s+ it):
U(x+ s, y + t) U(x, y) = Ux
s+U
yt+ |h|,
V (x+ s, y + t) V (x, y) = Vx
s+V
yt+ |h|,
(3)
, 0 h 0.
f(z) = f(z + h) f(z) = Ux
s+U
yt+ i(
V
xs+
V
yt) + |h|,
= + i. -, -
f(z) = (U
x iV
x)h+ |h|.
limh0
f(z)
h=U
x+ i
V
x, .. f (z) .
. - f (z)
f (z) =U
x+ i
V
x=V
y iU
y=U
x iU
y=V
y+ i
V
x.
4.1
, - :
1. f(z) g(z) z, ,, ( g(z) 6= 0) :
(f g) = f g, (fg) = f g + fg,( fg
)=f g fg
g2.
2. (z) z, f(w) w = (z), F (z) = f((z)) - z,
F (z) = f ((z)) (z).
23
-
5. f(z) = zm, m < 0 - , - C, z = 0 (zm) = mzm1. ,(z1) = (1z)
= 1z2 .. 2 (zm) = mzm1. ,
,
(zm) =( 1zm
)= mz
m1
z2m= mzm1.
6. ez C.. ez = ex(cos y + i sin y). U(x, y) = ex cos y,
V (x, y) = ex sin y. U(x, y), V (x, y) , - (x, y), -. w = ez
(ez) = (ex cos y)x + i(ex sin y)x = e
x(cos y + i sin y) = ez.
7. cos z, sin z, ch z, sh z C,
(sin z) = cos z, (cos z) = sin z,(sh z) = ch z, (ch z) = sh z.
.
(sin z) =(eiz eiz
2i
)=
1
2i
(eiz i eiz (i)) = eiz + eiz
2= cos z.
. 8. ln z = ln |z|+ i arg z - Ln z =
= ln |z|+ i arg z + 2kpii, k Z. (ln z) = 1z.
. z = rei, f(z) = U(r, ) + iV (r, ). - x = r cos, y = r sin -
U
r=
1
r V
,V
r= 1
r U
()
:
f (z) =r
z
(U
r+ i
V
r
)=
1
z
(V
iU
).
24
-
ln z = ln r + i, 0 < < 2pi ()
(ln z) =r
z((ln r)r + ()
r) =
1
z
-, , , - :
98. .) w = zz. .) w = z. .) w = z2z. .) w = zez.
.) w = |z|z. .) w = ez2. .) |z|Re z. .) w = sin 3z i.99. .) w = zRe z. .) w = z Im z. .) w = |z| Im z. .) w = ch z.100. , w = f(z)
, .101. , f(z) = U(x, y)+ iV (x, y)
D,
U
x Vx
+U
y Vy
= 0.
5.
5.1 . .
f(z), G. - - , z0 z, G. n z0, z1, . . . , zn1, zn = Z, . zk, zk+1 k. = max
k|zk+1 zk|.
. 0
limn1k=0
f(k)(zk+1 zk),
k, f(z)
f(z)dz.
25
-
. - - , f(z) = f(x + iy) == U(x, y) + iV (x, y) - - ,
f(z)dz ,
f(z)dz =
U(x, y)dx V (x, y)dy + i
V (x, y)dx+ U(x, y)dy,
.. f(z)
Udx V dy,
V dx+ Udy.
. f(z) = f(x + iy) = U(x, y) + iV (x, y), zk = xk + iyk,k = x
k + iy
k, zk = xk + iyk, xk = xk+1 xk, yk = yk+1 yk,
f(k) = Uk + iVk, Uk = U(xk, yk), Vk = V (x
k, yk).
,
n1k=0
f(k)zk =n1k=0
(Ukxk Vkyk) + in1k=0
(Vkxk + Ukyk).
. , , ,
f(z)dz =
U(x, y)dx V (x, y)dy + i
V (x, y)dx+ U(x, y)dy.
.
1.)
(af(z) + bg(z))dz = a
f(z)dz + b
g(z)dz, a, b C.2.)
1+2
f(z)dz =1
f(z)dz +2
f(z)dz
1+2 , 1 2 , 1 2.
3.)
f(z)dz = -
f(z)dz
- .
, .
26
-
4.) |
f(z)dz|
|f(z)|dz M l, M = sup
z|f(z)|, l - .
. :
|n1k=0
f(k)zk| n1k=0
|f(k)| |zk| n1k=0
|f(k)|sk,
zk = zk+1 zk, sk - zk zk+1. , -
4.). 1. z = z(t),
t [, ] ( z(t) = x(t) + iy(t)),
f(z)dz =
f(z(t))z(t)dt.
.
f(z)dz=
(U(z(t))x(t)V (z(t))y(t))dt+i
(V (z(t))x(t)+U(z(t))y(t)
)dt=
=
(U(z(t)) + iV (z(t))x(t) + i(U(z(t)) + iV (z(t)))y(t)
)dt =
f(z(t))z(t)dt.
2. f(z(t))z(t) = R(t) + iI(t),
f(z)dz =
R(t)dt+ i
I(t)dt,
..
f(z)dz - .
. - - , - z0 Z.
zndz =1
n+ 1
(Zn+1 zn+10
),
n 6= 1 ( n z = 0).
27
-
. z = z(t), t [, ], .
zndz =
zn(t)z(t)dt =1
n+ 1
d(z(t))n+1 =1
n+ 1(Zn+1 zn+10 ).
,
zndz . -
, - , Z = z0
zndz = 0.
5.2
1. G C , - , G, - G. , , .
.1.) {z : |z a| < r} - .2.) {z : |z a| r}, {z : r < |z a| < R} - .
012
3
.
0,1, . . . ,n , 1, . . . ,n - 0 (. .).
2. , -, 0 1, . . . ,n, (n+1)- .
. 4- -.
1 ( ). f(z) - G,
f(z)dz = 0, -
, G.. 1 .) 4 -
. 4
f(z)dz = M.
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- -
-
.
, M = 0. 4 41, . . . ,44. (. .)
4
f(z)dz =
41
f(z)dz + +44
f(z)dz (1)
28
-
4f(z)dz
= M , (1) k=1,2,3,4 , 4kf(z)dz
M4. , , k=1, ..
41
f(z)dz M
4.
41 - 4(2)1 , . . . ,4(2)4 - 4(2) ,
4(2)f(z)dz
M42.
, . 4 = 4(0), 41 = 4(1),4(2), . . . ,4(n), . . . ,
4(n)f(z)dz
M4n, n = 0, 1, 2, . . . . (2)
l - 4 = 4(0). 4(1),4(2), . . . ,4(n), . . . l
2,l
22, . . . ,
l
2n. . . .
, 4(n)
f(z)dz. -
, n +, z0, - . G, f(z) , .. f(z) - z0 . , > 0, > 0, z (z0 , z0 + ) : f(z) f(z0)
z z0 f(z0)
< . |z z0| < ,
|f(z) f(z0) (z z0)f (z0)| < |z z0|. (3), N N , n > N 4(n)
|z z0| <
29
-
4(n)
f(z)dz (3). , 4(n)
f(z)dz =
4(n)
(f(z) f(z0) (z z0)f (z0))dz,
4(n)
dz = 0 4(n)
zdz = 0. , (3),
4(n)
f(z)dz <
4(n)|z z0|dz. (4)
z 4(n), |z z0| < l2n
(4) 4(n)
f(z)dz < l
2n l
2n= l
2
4n.
(2) , M
4n<
l2
4n M < l2.
- , M = 0. 1 .2 .) - . , 1 .,
f(z)dz = 0.
- , - .
.
A
B
C
D
E
PPPPPPPP
JJJJ
ZZLLLL
(. .)
f(z)dz - , , :
f(z)dz =
ABCA
f(z)dz +
ACDA
f(z)dz +
ADEA
f(z)dz.
, -
, . 1 . ( - -)
f(z)dz = 0.
3 .) - - . ,
f(z)dz = 0. , > 0 -
P , ,
f(z)dz P
f(z)dz < . (5)
30
-
2 .P
f(z)dz = 0, (5) ,
f(z)dz = 0.
G f(z) G. D - G . f(z) - G. f(z) - D. f(z) - D.
> 0 > 0 z, z D : |z z| < |f(z) f(z)| < . n s0, s1, . . . , sn1, .
P , l0, l1, . . . , ln1 ., lk < , k = 0, . . . , n.
S =n1k=1
f(zk)zk =n1k=1
f(zk)
Sk
dz =n1k=0
Sk
f(zk)dz. (6)
:
f(z)dz =n1k=0
Sk
f(z)dz. (7)
(6) (7),
f(z)dz S =n1k=0
Sk
(f(z) f(zk))dz.
Sk : |f(z) f(zk)| < .
f(z)dz S < n1
k=0
|Sk| = l, (8)
l - , |Sk| - Sk. ,
P
f(z)dz S . S = n1
k=0
lk
f(zk)dz,
zk =lk
dz.P
f(z)dz S =n1k=0
lk
(f(z) f(zk))dz. lk : |f(z) f(zk)| < ,
P
f(z)dz S < n1
k=0
lk < l. (9)
(8) (9)
f(z)dz P
f(z)dz
f(z)dz S+ S
P
f(z)dz < 2l.
31
-
3 . 2 ( ( )). -
- f(z) - .
f(z)dz = 0.
. D - , , . z D , f(z) . D - -, , D. , , G, - . G f(z) G,
f(z)dz.
1. - , .
. - (n+1)- :
= +0 + 1 + + n .
0,1, . . . ,n - - , - 0, . . . ,n 0.
q
q
q
q
q
q
b
b
a
c
d
l
l
n
m
f e
e
2+0
1
,
=
+0
+
1
+ +n
= 0.
= abcdefmlna, =anlmfedcba.
f(z)dz = 0,
f(z)dz = 0.
an, mf ,dc - .
f(z)dz =
f(z)dz +
f(z)dz = 0.
k
f(z)dz = k
f(z)dz,
0
f(z)dz =
1
f(z)dz + +n
f(z)dz.
32
-
2. , f(z) G z0, z G,
f(z)dz, - , G,
z0 z, , z0 z..
1+
2
f(z)dz = 0, ..
G
qq
z0
z1
2
1
f(z)dz +
2
f(z)dz = 0.
1
f(z)dz =
2
f(z)dz.
5.3
. f(z), G, (z), G: (z) = f(z).
f(z) G. F (z) =zz0
f()d,
z0, z G - - , G, z0 z. : F (z) , , F (z) , G.
1. f(z) G, F (z) f(z).
. , z G : F (z) = f(z). z + h G.
F (z + h) F (z) =z+hz0
f()d z
z0
f()d =
z+hz
f()d. (1)
z+hz
d = h,
f(z) =f(z)
h
z+hz
d =1
h
z+hz
f(z)d. (2)
33
-
(1) (2) , , z z + h.
(1) (2),
F (z + h) F (z)h
f(z) = 1h
z+hz
(f() f(z))d.
f(z) - G f(z) - z G,.. > 0 > 0 , |z| < , |f()f(z)| < . f(z), z G, F (z + h) F (z)h f(z)
< |h||h| = |h| < . lim
h0F (z + h) F (z)
h= f(z), .. F (z) = f(z).
2. f(z) - G. f(z) :
(z) = F (z) + C, F (z) =z
z0
f()d, C - const, z0, z G.
. (z) , (z) = f(z). 1: F (z) = f(z). (z) F (z) = ((z) F (z)) = f(z) f(z) = 0. (z) F (z) = (z) = U(x, y) + iV (x, y). (z) = U x + iV x =V y iU y. (z) = 0, U x = V x = V y = U y = 0 G. U V - onst G (z) = U + iV - const (z) = F (z) + C.
1. 2 -:
z1z0
f()d = (z1) (z0).
. z = z0 (z) =zz0
f()d + C.
C = (z0). , , z = z1,
(z1) =
z1z0
f()d + (z0),
34
-
- . 2. f(z) g(z)
G, z0 z1 - ,
z1z0
f(z)g(z)dz = f(z)g(z)|z1z0 z1z0
g(z)f (z)dz.
.
(f(z)g(z)) = f (z)g(z) + f(z)g(z) :
z1z0
(f(z)g(z))dz =
z1z0
f (z)g(z)dz +
z1z0
f(z)g(z)dz.
- z1z0
(f(z)g(z))dz = f(z1)g(z1) f(z0)g(z0) = f(z)g(z)z1z0.
.
. - -.
z = (w) - 1 w- z-.
f(z)dz =
1
f((z))(z)dz.
1.
(1 + i 2z)dz
, z1 = 0 z2 = 1 + i:1.) ;2.) y = x2;
35
-
3.) z1z3z2, z3 = 1.. 1 + i 2z = (1 2x) + i(1 + 2y). U(x, y) = 1 2x,
V (x, y) = 1 + 2y.
f(z)dz =
U(x, y)dx V (x, y)dy + i
V (x, y)dx+ U(x, y)dy,
(1 + i 2z)dz =
(1 2x)dx (1 + 2y)dy + i
(1 + 2y)dx+ (1 2x)dy.
1.) , z1 = 0 z2 = 1 + i, y = x, x [0 ;1].
(1+i2z)dz =1
0
((12x)+(1+2x))dx+i1
0
((1+2x)+(12x))dx = 2(i1).
2.) y = x2 dy = 2xdx, x [0 ;1].
(1+ i2z)dz =1
0
(12x (1+2x2) 2x)dx+ i1
0
(1+2x2 +(12x) 2x)dx =
= 2 + 43i.
3.) z1z3: y = 0, dy = 0, 0 x 1. z2z3: x = 1,dx = 0, 0 y 1.
(1 + i 2z)dz =z1z3
(1 + i 2z)dz +z3z2
(1 + i 2z)dz =
=
10
(1 2x)dx+ i1
0
dx1
0
(1 + 2y)dy +
10
(1 2 1)dy = 2.
2.
(z2 + zz)dz,
- |z| = 1, 0 arg z pi.
36
-
. z = ei, 0 pi. dz = ieid
(z2 + zz)dz =
pi0
iei(ei2 + 1)d = i
pi0
(ei3 + ei)d =
(1
3ei3+ei
)pi0
= 83.
3.
ezdz, - y = x, z1 = 0 z2 = pi ipi.
. : x = t,y = t, : z = t it, 0 t pi.
ez =
pi0
et+it(1 i)dt = (1 i)pi
0
e(1+i)tdt =1 i1 + i
e(1+i)tpi0
= (epi + 1)i.
4.
2+i1i
(3z2 + 2z)dz.
. f(z) = 3z2 + 2z C, - 2+i
1i(3z2 + 2z)dz= (z3+z2)
2+i1i
= (2 + i)3 + (2 + i)2 (1 i)3 (1 i)2 = 8 + 19i.
5.
i0
z cos zdz.
. f(z) = z, g(z) = cos z C, :
i0
z cos zdz =
i0
z(sin z)dz = z sin zi0
i
0
sin zdz = i sin i+ cos z
i0
=
= iei
2 ei22i
+ei
2
+ ei2
2 1 = sh 1 + ch 1 1 = 1 e
e.
37
-
:102.
z Im z2dz, : |z| = 1 (pi arg z 0).103.
e|z|2
Re zdz, - , z1 = 0, z2 = 1+i.
104.
ln zdz (ln z = ln |z|+ i arg z - ), : |z| = 1, ) z0 = 1; ) z0 = 1. .
105.
zRe zdz, : |z| = 1. .106.
zzdz, : |z| = 1. .
107.i
1
zezdz.
108.
Re zdz, : ) z = (2 + i)t, 0 t 1; ) , [0 ; 2] , z1 = 2 z2 = 2 + i.
109.1i1+i
(2z + 1)dz.
110.i+10
z3dz.
111.i
1
(3z4 2z3)dz.112.
ezdz, : ) y = x2, z1 = 0
z2 = 1 + i; ) , .113.
cos zdz, : , z1 =pi
2
z2 = pi + i.
114.2i
1+i
(z3 z)e z22 dz.
115.i
0
z cos zdz.
116.i
1
z sin zdz.
117.i
0
(z i)ezdz.
38
-
118.i
1
ln (z+1)z+1 dz |z| = 1, Im z 0, Re z 0.
119.i
1
ln zz dz , z1 = 1 z2 = i.
120.1+i0
sin z cos zdz.
121.i
1
1+tg zcos2 z dz , z1 = 1 z2 = i.
122.
Re (sin z) cos zdz, : | Im z| 1, Re z = pi4 .123.
z Im z2dz, : | Im z| 1, Re z = 1.
124.iizez
2
dz.
125.
tg zdz, : y = x2, z1 = 0
z2 = 1 + i.
126.ii|z|dz : ) ; ) -
|z| = 1; ) |z| = 1.127.
1+i0
ezdz : ) 0, 1, 1 + i; )
0, i, 1 + i.
5.4
( ). G - n- , - - f(z) - - G .
f(z) =1
2pii
f()
zd,
z G ,.. G .
. (n+ 1)- G = G \S, S - r z G. r , S G. - S, . :
f()
zd +-
f()
zd = 0.
39
-
f()
zd =
f()
zd. (1)
: = z + rei, d = ireid,
d
z =2pi
0
ireid
rei= 2pii. f(z) = 1
2pii
f(z)
zd. (2)
(1) (2) :
1
2pii
f()
zd f(z) =1
2pii
f()
zd 1
2pii
f(z)
zd =
=1
2pii
f() f(z) z d. (3)
(3): 12pii
f() f(z) z d
12pi max |f() f(z)|2pi = max |f() f(z)| . f(z) z, r 0
max|f() f(z)| 0. 1
2pii
f() f(z) z d = 0.
1
2pii
f()
zd = f(z).
.. , , -
|z| = R z R, , z = Rei,
f(z) =1
2pi
2pi0
f(z +Rei)d. (4)
(4) . ( ). f(z) ,
.
40
-
1. |z|=2
ch iz
z2 + 4z + 3dz.
. z0 = 1, z2 + 4z+ 3 = 0, |z| = 2. f(z).
|z|=2
ch iz
z2 + 4z + 3dz =
|z|=2
ch iz
(z + 1)(z + 3)dz =
|z|=2
ch izz+3
z (1)dz.
, z0 = 1, f(z) = ch izz+3 . f(z) - |z| 2. ,
|z|=2
ch iz
z2 + 4z + 3dz = 2piif(1) = 2piich (i)
2= pii ch i = pii cos 1.
2. ,
ez2
z2 6zdz,
1.) : |z 2| = 1; 2.) : |z 2| = 3; 3.) : |z 2| = 5.. 1.)
ez2
z2 6z - |z 2| = 1. , :
|z2|=1
ez2
z2 6zdz = 0.
2.) z2 6z = z(z 6) = 0 z = 0 z = 6. z = 0 |z 2| = 3, f(z) = ez2z6 - |z 2| 3. ,
|z2|=3
ez2
z2 6zdz =
|z2|=3
ez2
z6zdz = 2piif(0) = 2pii
( 1
6
)= pi
3.
41
-
3.) z = 0 z = 6 z2 6z - |z 3| = 5. . .
I. 1
z2 6z
1
z2 6z =1
6 1z 6
1
61z.
|z2|=5
ez2
z2 6zdz =1
6
|z2|=5
ez2
z 6dz 1
6
|z2|=5
ez2
zdz =
=1
62piiez
2z=6 1
62piiez
2z=0
=e36 1
3pii.
II. z = 0 z = 6 1 2, |z 2| 5 (. ). , |z2| = 5, 1 2, .
|z2|=5
ez2
z2 6zdz =1
ez2
z2 6zdz +2
ez2
z2 6zdz.
6
-
qqq qq
3 0 2 6 7x
y
1 2
.
- . -
|z2|=5
ez2
z2 6zdz = 2piiez
2
z 6z=0
+
+ 2piiez
2
z
z=6
=e36 1
3pii.
C :
42
-
128.
|z|=1
ez
z2 + 2zdz. 129.
|zi|=1
eiz
z2 + 1dz.
130.
|z1|=2
sin piz2z2 + 2z 3dz. 131.
|z|=2
sin iz
z2 4z + 3dz.
132.
|z|=1
tg z
ze1z+2
dz. 133.
|z|=3
cos (z + pii)
z(ez + 2)dz.
134.
|z|=5
dz
z2 + 16. 135.
|z|=4
dz
(z2 + 9)(z + 9).
136.
|z|=1
sh pi2 (z + i)
z2 2z dz. 137.|z|=2
sin z sin (z 1)z2 z dz.
5.5 .
1. f(z) G , n
f (n)(z) =n!
2pii
f()
( z)n+1d,
G.. .
n = 1. ,
f (z) = limh0
f(z+h) f(z)h
=1
2piilimh0
1
h
f()
(1
z h 1
z)d =
=1
2piilimh0
f()
( z h)( z) =1
2pii
f()
( z)2d.
, h 0
1
z h 1
z . f (n)(z) n m 1.
f (m)(z) =(f (m1)(z)
)= lim
h0f (m1)(z + h) f (m1)(z)
h=
=(m 1)!
2piilimh0
1
h
f()
(1
( z h)m 1
( z)m)dz. (1)
43
-
:
1
( z h)m 1
( z)m =( z)m ( z h)m
( z)m( z h)m =
=( z)m( z)m(1 hz)m
( z)m( z h)m =( z)m( z)m(1 mhz+ O (h2))
( z)m( z h)m =
=mh( z)m1 + O (h2)( z)m( z h)m =
mh
( z)( z h)m +O (h2)
( z)m( z h)m . (1):
f (m)(z) =m!
2piilimh0
f()d
(z)(zh)m +(m1)!
2piilimh0
f() O (h)d
(z)m(zh)m =
=m!
2pii
f()
( z)m+1d.
. f (n)(z) - z. - .
2 ( ). f(z) - |z z0| R,
|f (n)(z0)| n!MRn
,
M = maxz|f(z)|, = {z : |z z0| = R}.
. 1
f (n)(z0) =n!
2pii
f()
( z0)n+1d.
|f (n)(z0)| n!M2piRn+1
dl =n!M 2piR
2piRn+1=n!M
Rn.
5.6 .
(. ). f(z) C , .
. z C : |f(z)| M . R > 0 , |f (z)| M
R.
44
-
R +, |f (z)| 0. - ,
f(z) f(z0) =z
z0
f (z)dz = 0. f(z) const.
(. ). f(z) G
f(z)dz G , f(z) G.
. , zz0
f()d
, .. z0 -
F (z) =zz0
f()d. , F (z) = f(z),
f(z) G. , - f(z) G. ,F (z) = f(z) z G, .. F (z) G. , , f(z), - , . .
1. |z1|=1
sin piz
(z2 1)2dz.
. sinpiz(z21)2 |z1| 1, z0 = 1.
sin piz
(z2 1)2 =sinpiz(z+1)2
(z 1)2 .
, f(z) = sinpiz(z+1)2 |z 1| 1. f (n)(z) n = 1.
|z1|=1
sinpiz
(z2 1)2dz =
|z1|=1
sinpiz(z+1)2
(z 1)2dz = 2piif(1).
45
-
f (z) =(
sin piz
(z+1)2
)=pi cos piz(z+1)2 sinpiz
(z+1)3. f (1) = 2pi cospi
23= pi
4.
, |z1|=1
sin piz
(z2 1)2dz = pi2
2i.
2. |z|=2
ch z
(z + 1)3(z 1)dz.
. I. 1
(z + 1)3(z 1) -
1
(z + 1)3(z 1) =1
8 1z 1
1
8 1z + 1
14 1(z + 1)2
12 1(z + 1)3
.
, |z|=2
ch z
(z + 1)3(z 1)dz =1
8
|z|=2
ch z
z 1dz 1
8
|z|=2
ch z
z + 1dz
14
|z|=2
ch z
(z + 1)2dz 1
2
|z|=2
ch z
(z + 1)3dz.
:|z|=2
ch z
z 1dz = 2pii ch 1,|z|=2
ch z
z + 1dz = 2pii ch 1.
- :
|z|=2
ch z
(z + 1)2dz = 2pii(ch z)
z=1
= 2pii sh 1,|z|=2
ch z
(z + 1)3dz =
2pii
2!(ch z)
z=1
= pii ch 1.
46
-
|z|=2
ch z
(z + 1)3(z 1)dz =2pii ch 1
8 2pii ch 1
8+
2pii sh 1
4 pii ch 1
2=
=sh 1 ch 1
2pii = pii
2e.
II. (z + 3)3(z 1) = 0 z = 1 z = 1. 1 2 , 1 2 |z| 2. , |z| = 2,1 2,
ch z
(z + 3)3(z 1) . , :
|z|=2
ch z
(z + 1)3(z 1)dz =1
ch z
(z + 1)3(z 1)dz +2
ch z
(z + 1)3(z 1)dz.
ch z
(z + 1)3(z 1) =ch zz1
(z + 1)3.
ch zz1 - 1, , ,
1
ch zdz
(z + 1)3(z 1) =1
ch z(z1)
(z + 1)3dz =
2pii
2!
(ch z
z 1)
z=1= pii
2e1 + ch 14
.
:2
ch z
(z + 1)3(z 1)dz =2
ch z(z+1)3
(z 1)dz = 2piich z
(z + 1)3
z=1
= piich 1
4.
:|z|=2
ch z
(z + 1)3(z 1)dz = pii2e1 + ch 1
4+ pii
ch 1
4= pii
2e.
47
-
138.
|z|=1
cos z
z2dz. 139.
|z|=1
sh2 z
z3dz.
140.
|z1|=1
sin pi4z
(z 1)2(z 3)dz. 141.|z|=2
z sh z
(z2 1)2dz.
142.
|z3|=6
z dz
(z 2)3(z + 4) . 143.
|z2|=3
ch eipiz
z3 4z2dz.
144.
|z|= 12
1
z3cos
z
z + 1dz. 145.
|z2|=1
e1z
(z2 + 4)2dz.
146.
|z|= 12
1 sin zz2
dz. 147.
|z1|= 12
eiz
(z2 1)2dz.
6.
6.1
qaqz
G
q C , |q| < 1, n :
1
1 q = 1 + q + q2 + + qn + q
n+1
1 q (1)
f(z) - G, z, a - G. - - , G, z a. (1) :
1
z =1
a 1
1 zaa=
1
a
(1 +
z a a +
(z a a
)2+ +
+
(z a a
)n+
1
1 zaa(z a a
)n+1).
48
-
f()
2pii
. ,
f(z) = f(a) +f (a)
1!(z a) + + f
(n)(a)
n!(z a)n +Rn,
Rn :
Rn =(z a)n+1
2pii
f()d
( z)( a)n+1 .
: Rn 0 n , , f(z) a, ..
f(z) =n=0
f (n)(a)
n!(z a)n. (2)
. (. ). f(z)
(2) a, . , , (1) .
. f(z) |z a| < R. R 0 < R < R; = {z :|z a| = R}; z {z : |z a| < kR}, 0 < k < 1. | z| R kR = (1 k)R , ,
|Rn| = (z a)n+1
2pii
f()d
( z)( a)n+1 kn+1(R)n+1
2pi M2piR
(1 k)(R)n+2 =
=Mkn+1
1 k , M = max|za|
-
6.2
1 ( ). n=1
fn(z), -
, G, - , S(z) - G.
. , S(z) - , , - G. G.
S(z)dz =n=0
fn(z)dz = 0,
fn(z)dz = 0 n 0., , , S(z)
G. 2 ( ).
n=0
fn(z), -
G G, - G, G .
. - G z G. min| z| = m > 0,
n=0
fn(z)(z)k+1
k, , , , :
S(k)(z) =k!
2pii
S()d
( z)k+1 =n=0
k!
2pii
fn()d
( z)k+1 =n=0
f (k)n (z).
, - .
2 .
:n=0
an (za)n, an - . -, , , - . , , - -
50
-
n=0
an(z a)n: 1R = limnn|an|, ,
, |z a| < R, - . , :
3 ( ). - .
. z, , G, . , - G fn(z) = an(z a)n G.
6.3 ,
1. -.
. |za|r n=0
an(za)n== f(z) . z = a, a0 = f(a). z = a, - n 1 :
f (n)(a) = n! an an = f(n)(a)
n!.
. f(z) a, - f(a) = 0. f(z) 6 0 , ( a) 6= 0. 6= 0 - a. , n :
f(z) =k=n
ak(z a)k, n > 1, an 6= 0. ()
. , f (n)(z).
2 ( ). f(z) a 6 0 . z = a, f(z) a.
. a - n. (*)
f(z) = (z a)n(z), (z) =k=0
an+k(z a)k, (a) = an 6= 0.
51
-
(z) a, . (z) - . (z) 6= 0 a, .. f(z) 6= 0 . .
3 ( ). f1(z) f2(z) G an a G, G f1(z) = f2(z).
. f(z) = f1(z) f2(z). - G an. f - an a, f(an) f(a) = 0. , f(z) = 0 - a, 2. , f(z) . E - f(z). E = G, . E 6= G. b (G) (bn) E , bn b. f(z) -, f(bn) f(b) = 0. f 6 0 b, b f ., 2 ( ), b f(z). , b = lim bn. , E = G.
1.
n=1
ein
n2.
. , n=1
zn, zn = xn + iyn,
, n=1
xn,n=1
yn. ein = cosn+ i sinn.
n=1
cosn
n2,n=1
sinn
n2 ,
(n=1
zn , n=1|zn| ).
52
-
2.
n=1
eipin
n.
. eipin = cos pin + i sin
pin .
n=1
cos pinn
, n=1
sin pinn
. , .
:
148.n=1
cos in
2n. 149.
n=1
n sin in
3n.
150.n=1
cos in2
5n2. 151.
n=1
ei2n
nn.
152.n=1
eipinn. 153.
n=1
(1 + i)n
2n2 cos in
.
154.n=1
sh in
sin in. 155.
n=1
lnn
sh in.
156.n=1
ch ipinnlnn
. 157.n=1
n
tg ipin.
1.
n=0
cos in zn.
. an = cos in =en + en
2= chn. -
:
R = lim|an||an+1| . R = lim
| chn|| ch (n+ 1)| = lim
chn
chn ch 1 + shn sh 1 =
53
-
= lim1
ch 1 + thn sh 1 = lim1
ch 1 + sh 1= e1. , -
R = e1.
2. n=1
(1 + i)nzn.
. an = (1 + i)n. |an| = |1 + i|n =(
2)n
= 2n2 . R =
= lim1
n|an| = lim 1n2n2 = 12 .
:
158.n=1
eizzn. 159.n=1
eipinzn.
160.n=0
( z1 i
)n. 161.
n=1
( zin
)n.
162.n=1
chi
nzn. 163.
n=1
( zln in
)n.
164.n=0
inzn. 165.n=1
sinpii
n zn.
166.n=1
cosnpiinzn. 167.
n=1
zn
sinn (1 + in).
168.n=0
(n+ 1)zn. 169.n=0
cos in zn.
f(z), z = z0,
f(z) =k=0
ck(z z0)k,
ck =1
2pii
f(z)dz
(z z0)k+1 =f (k)(z0)
k!, k = 0, 1, 2, ... , - -
z = z0, z0, f(z) .
54
-
ez,sin z, cos z, sh z, ch z, ln (1 + z), (1+z) z0 = 0:
ez =k=0
zk
k!(R = +), sin z =
k=0
(1)kz2k+1(2k + 1)!
(R = +),
cos z =k=0
(1)kz2k(2k)!
(R = +), sh z =k=0
z2k+1
(2k + 1)!(R = +),
ch z =k=0
z2k
(2k)!(R = +), ln (1 + z) =
k=1
(1)k1zkk
(R = 1),
(1 + z) =k=0
( 1)...( k + 1)k!
zk (R = 1).
, = 1 1
1 + z=
k=0
(1)kzk (R = 1), 11 z =
k=0
zk (R = 1).
R - .
1.
f(z) =z
z2 2z 3 .
.z
z2 2z 3 =1
4 1z + 1
+3
4 1z 3 =
1
4 1
1 + z 1
4 1
1 z3.
1
1 + z
1
1 z ,
f(z) =1
4
(1 z + z2 z3 + . . .
) 1
4
(1 +
z
3+z2
9+ . . .
)=
=1
4
( 4
3z +
8
9z2 28
27z3 + . . .
)= z
3+
2
32z2 7
33z3 + . . . .
z0 = 0 z0 = 1. R = 1.
2. f(z) =1
3 2z - z = 3, .. z 3.
.1
3 2z =1
3 2(z3+3) =1
3 2(z3) = 1
3 1
1 + 23(z3).
55
-
1
1 + z, z
23(z 3).
1
3 2z = 1
3,(
1 23
(z 3) + 22
32(z 3)2 2
3
33(z 3)3 + . . .
)=
= 13
+2
32(z 3) 2
2
33(z 3)2 + 2
3
34(z 3)3 . . . .
2
3(z 3)
< 1. |z 3| < 32, ..
R =3
2.
, , :
170. sin (2z + 1) z + 1.
171. cos z z +pi
4.
172. ez 2z 1.173.
1
3z + 1 z + 2.
174.z + 1
z2 + 4z 5 z.175.
z
z2 + i z.
176. cos2iz
2 z.
177. sh2z
2 z.
178. ln (2 z) z.179. ln (2 + z z2) z. -
z . :
180.1
1 + ez. 181.
1
2 + sin z. 182.
1
ez + 5.
183. ln (1 + ez). 184. ln cos z. 185. ln (1 + cos z).
186. e1
1z .
56
-
f(z) z0. z0 - f(z) n, f(z0) = 0, . . . , f (n1)(z0) = 0,f (n)(z0) 6= 0. n = 1 z0 .
, z0 n- - f(z), f(z) = (zz0)n(z), (z) - z0 (z0) 6= 0.
1. f(z) = 1 + cos z .
. 1 + cos z = 0. zn = (2n+ 1)pi, n Z. f ((2n+ 1)pi) = sin ((2n+ 1)pi) = 0,f ((2n+ 1)pi) = cos ((2n+ 1)pi) = 1 6= 0.
, zn = (2n + 1)pi, n Z, f(z) = 1 + cos z.
2. f(z) = 1 ez -.
. 1 ez = 0. ez = 1. zn = 2pini (n Z) - f(z) = 1 ez. , f (zn) = e2pini = 1 6= 0. , zn =2pini (n Z) - ( ) f(z) = 1ez.
3. z0 = 0
f(z) =z8
z sin z .
. f(z) =z8
z (zz33! +z5
5! . . . )=
z8
z3
3!z5
5! + . . .=
z5
13!z
2
5! + . . .=
= z5 113! z
2
5! + . . ..
(z) =1
13! z
2
5! + . . .. f(z) = z5(z), (z) - -
z0 = 0, (0) = 6 6= 0. , z0 = 0 .
57
-
:
187. ) f(z) = z4 + 4z2; ) f(z) =sin z
z.
188. ) f(z) = z2 sin z; ) f(z) =sh2 z
z.
189. ) f(z) = 1 + ch z; ) f(z) =(1 sh z)2
z.
190. ) f(z) = (z + pii) sh z; ) f(z) = cos z3.191. ) f(z) = (z2 + pi2)(1 + ez); ) f(z) = cos z + ch iz. z0 = 0 :
192. f(z) =z6(
z2
)2 (sin z2)2 . 193. f(z) = esin z etg z.194. f(z) =
z3
1 + z ez . 195. f(z) = 2(ch z 1) z2.
196. f(z) =(1 cos 2z)2z sh z . 197. f(z) = (e
z ez2) ln (1 z).198. f(z) = z2(ez
2 1). 199. f(z) = 6 sin z3 + z3(z6 6).
7.
7.1
(. ). K : r < |z a| < R, 0 r r, .. z, |z a| = r.
1n=
cn(z a)n =n=1
cn(z a)n .
60
-
7.3
. f(z) =
n=cn(z a)n
r < |z a| < R.
. r < |z a| < R
f(z) =
n=cn(z a)n =
n=
cn(z a)n. (1)
k = 0,1,2, . . . . (1) (z a)k1.
ck(z a)1+
n=,n 6=k
cn(z a)nk1 = ck(z a)1+
n=,n 6=k
cn(z a)nk1. (2)
,
dzza = 2pii , -
a. a, r < |z a| < R. (2) - .
cn(za)nk1dz = cn
(az)nk1dz = 0 k 6= n, 2piick = 2piick,, ck = ck k = 0,1,2, . . . . .
1. 0 < |z1| < 2
f(z) =1
(z2 1)2 .
. I. f(z) = 1(z21)2 0 < |z 1| < 2. :
cn =1
2pii
1(z21)2
(z 1)n+1dz =1
2pii
dz
(z 1)n+3(z + 1)2 ,
- z0 = 1, .
61
-
n+3 0, .. n 3, 1(z1)n+3(z+1)2 , z = 1.
dz
(z 1)n+3(z + 1)2 = 0,
.. cn = 0 n = 3,4, . . . . n + 3 > 0, .. n > 3, , - ,
cn =1
2pii
1(z+1)2
(z 1)n+3dz =1
(n+ 2)!
dn+2
dzn+2
[1
(z + 1)2
] z=1
=
=1
(n+ 2)!
(1)n(n+ 3)!(z + 1)n+4
z=1
=(1)n(n+ 3)
2n+4.
, n = 2,1, 0, 1, . . .
cn =(1)n(n+ 3)
2n+4.
f(z) = 1(z21)2 0 < |z 1| < 2
1
(z2 1)2 =+n=2
cn(z 1)n =+n=2
(1)n(n+ 3)2n+4
(z 1)n,
1
(z2 1)2 =1
4 1
(z 1)2 1
4 1z 1 +
3
16 1
8(z 1)+
+5
64(z 1)2 3
64(z 1)3 + . . . .
II. :
f(z) =1
(z2 1)2 =1
4
( 1z 1
1
z + 1
)2=
=1
4 1
(z 1)2 1
4 1z 1 +
1
4 1z + 1
+1
4 1
(z + 1)2. (1)
1
z + 1=
1
(z 1) + 2 =1
2 1
1 + z12,
1
(z + 1)2=
1
4(
1 +( z 1
2
) )2.
62
-
z (1 + z) =1 = 2,
1
z + 1=
1
2
(1 z 1
2+( z 1
2
)2( z 1
2
)3+ . . .
),
1
(z + 1)2=
1
4
(1 2 z 1
2+2(2 1)
2!( z 1
2
)2
2(2 1)(2 2)3!
( z 1
2
)3+ . . .
).
(1),
1
(z2 1)2 =1
4 1
(z 1)2 1
4 1z 1 +
1
8
(1 z 1
2+( z 1
2
)2
( z 1
2
)3+ . . .
)+
1
16
(1 (z 1) + 3
22(z 1)2 4
23(z 1)3 + . . .
),
1
(z2 1)2 =1
4 1
(z 1)2 1
4 1z 1 +
3
16 1
8(z 1)+
+5
64(z 1)2 3
64(z 1)3 + . . . .
2.
f(z) =2z + 1
z2 + z 2 ,
a = 0.. z2 + z 2 = 0 z1 = 2 z2 = 1. ,
a = 0, f(z) :
.) |z| < 1;.) 1 < |z| < 2;.) |z| > 2 - |z| 2.
f(z) . f(z)
f(z) =1
z + 2+
1
z 1 . (1)
.) f(z) |z| < 1.
63
-
(1) :
f(z) =1
z 1 +1
z + 2=
1
2 1
1 + z2 1
1 z . (2)
(1 + z) = 1, 1
1 z = 1 + z + z2 + . . . , |z| < 1, (3)
1
1 + z2= 1 z
2+z2
4 z
3
8+ . . . , |z| < 2. (4)
(3) (4) (2),
2z + 1
z2 + z + 1=
1
2 z
4+z2
8 z
3
16+ (1 + z + z2 + . . . ) =
= 12 3
4z 7
8z2 15
16z3 + . . . .
f(z)..) f(z) 1 < |z| < 2. (4)
1
1 + z2 ,
|z| < 2. (3) 11 z |z| > 1.
f(z) :
f(z) =1
2 1
1 + z2+
1
z 1
1 1z. (5)
(3),
1
1 1z= 1 +
1
z+
1
z2+
1
z3+ . . . . (6)
(4) (6) (5),
2z + 1
z2 + z 2 =1
2 z
4+z2
8 z
3
16+ + 1
z+
1
z2+ =
= + 1z2
+1
z+
1
2 z
4+z2
8 z
3
16+ . . .
2z + 1
z2 + z 2 =n=1
1
zn+
1
2
n=0
zn
2n.
64
-
.) f(z) |z| > 2. (4)
1
1 + z2 |z| > 2 , (6)
1
1 1z , , |z| > 2, |z| > 1.
f(z) :
f(z) =1
z 1
1 + 2z+
1
z 1
1 1z.
, (3) (4),
f(z) =1
z
(1 2
z+
4
z2 8z3
+ + 1 + 1z
+1
z2+
1
z3+ . . .
)=
=1
z
(2 1
z+
5
z2 7z3
+ . . .)
2z + 1
z2 + z 2 =2
z 1z2
+5
z3 7z4
+ . . . .
, , f(z) .
3.
f(z) =2z 3
z2 3z + 2 z1 = 1 z2 = 2, z2 3z + 2 = 0.
. 1.) f(z) z1 = 1, .. 0 < |z 1| < 1.
f(z)
2z 3z2 3z + 2 =
1
z 1 +1
z 2 .
:
2z 3z2 3z + 2 =
1
z 1 1
1 (z 1) .
, 1
1 (z 1) z 1,
2z 3z2 3z + 2 =
1
z 1 (1 + (z 1) + (z 1)2 + . . . )
65
-
2z 3
z2 3z + 2 =1
z 1 n=0
(z 1)n.
2.) f(z) z2 = 2, .. 0 R} - z =.
1. z = - f(z), U(), f(z).
z = - - f(z). (z) = f
( 1z
): z =
1
z. ,
(z) z - (z).
2. f(z) , ( n), -
69
-
, z = 0 , (- n), (z), .
f(z) U() = {z : |z| > R}. (z) =f( 1z
), z =
1
z. (z) |z| < 1
R
z = 0. (z) :
(z) =
n=bn(z
)n. (1)
z =1
z,
f(z) =
n=bn
1
zn=
n=
bnzn =
n=cnz
n, cn = bn. (2)
(2) ez, z - , (1) - z, z, z, .
1. 1. f(z) , (2) z.
2. f(z) n, (2) z, cn 6= 0, ck = 0 k > n.
3. f(z) , (2) .
2. limz f(z) = c0. f(z) -
, f(z) . c0 f(z) , , .. (z) z = 0. , f() = c0 = lim
z f(z), , f(z) z =.
:
1. f(z) =ez 1z
.
70
-
. f(z) z = 0.
limz0
f(z) = limz0
ez 1z
= 1.
, z = 0 .
2. f(z) =1
z3.
. z = 0. z = ei, f(z)=ei3
3.
|f(z)| = 13. lim
z0|f(z)| = , .. z = 0 .
(z) = z3 z = 0 , -
, z = 0 f(z) =1
z3.
z0 n f(z), , f(z) -
f(z) =(z)
(z z0)n , (z) z0 (z0) 6= 0.
3. f(z) =sin z
z3 + z2 z 1 .. f(z) z = 1 z = 1.
z = 1. f(z)
f(z) =
sin z
z 1(z + 1)2
.
(z) =sin z
z 1 z = 1,
(1) = sin 126= 0. , z = 1
. , f(z)
f(z) =
sin z
(z + 1)2
z 1 ,, z = 1 .
4. z = 0 f(z) =e1/z
2..
. z = x f(x) = e1/x2 x 0. z = iy f(iy) = e1/y2 0 y 0. , - f(z) z = 0 . z = 0 f(z).
71
-
5. z = 0
f(z) =1
2 + z2 2 ch z .
. z = 0 f(z),
. (z) =1
f(z)= 2+z22 ch z.
(0) = 0. z = 0 . :
(z) = 2z 2 sh z, (0) = 0;(z) = 2 2 ch z, (0) = 0;(z) = 2 sh z, (0) = 0;IV (z) = 2 ch z, IV (0) = 2 6= 0.
z = 0 (z), , f(z) z = 0 .
6. z = 0
f(z) =1 ez
z.
. ez z0 = 0, f(z)
f(z) =1
z(1 ez) = 1
z
(1(
1 z + z2
2! z
3
3!+ . . .
))= 1 z
2!+z2
3! . . . .
. z0 = 0 .
f(z) , - z0 = 0.
7. z = 0
f(z) =1 cos z
z7.
. cos z z, - f(z) :
f(z) =1
z7
(z2
2! z
4
4!+z6
6! . . .
)=
1
2!z5 1
4!z3+
1
6!z . . . .
72
-
, z0 = 0 .
8. z = 1
f(z) = (z 1)e 1z1 .
. , f(z) z = 1:
f(z) = (z 1)(
1 +1
z 1 +1
2!(z 1)2 +1
3!(z 1)3 + . . .)
=
= 1 + (z 1) + 12!(z 1) +
1
3!(z 1)2 + . . . . -
z 1, z = 1 f(z).
z0 = 0 :
226. a)1
z sin z ; )1
cos z 1 + z22; )
1
ez + z 1 .
227. )sin z
ez + z 1; )ch z
z sh z . :
228. )1
1 sin z ; )1 cos z
z2.
229. ) e1z+2 ; ) cos
1
z.
230. )z
z5 + 2z4 + z3; )
1
ez 1 +1
z2.
231. ) e1z2 ; ) sin
pi
z + 1; ) ch
1
z.
232. )z2
cos z 1; )1 sin z
cos z; )
z pisin2 z
.
73
-
:
233.1 + cos z
z pi , z = pi. 234.z2 3z + 2z2 2z + 1 , z = 1.
235.sin z
z2, z = 0. 236.
sh z
z, z = 0.
237. cos1
z + pi, z = pi. 238. z
2 1z6 + 2z5 + z4
, z = 0, z = 1.
239.ln (1 + z3)
z2, z = 0. 240.
sin2 z
z, z = 0.
241.ez+e
z + e, z = e. 242. cos 1
z+ sin
2 piz2z
, z = 0.
243. z sh1
z, z = 0.
8.
8.1
. a - f(z). 12pii
C
f(z)dz f(z) -
a. C - , a, f(z) , a. :
res f(z)z=a
= res f(a) =1
2pii
C
f(z)dz.
f(z) :
f(z) =
n=cn(z a)n.
C , ,
C.C
f(z)dz =
n=cnC
(z a)ndz.
C
(z a)ndz = 0, n 6= 1 C
dz
z a = 2pii, C
f(z)dz = 2pii.
, res f(a) =1
2pii
C
f(z)dz = c1.
. z = a , c1 = 0 , -, res f(a) = 0.
74
-
( ). f(z) - G, a1, . . . , an G. C - - , -G a1, . . . , an. , C ,
C
f(z)dz = 2piink=1
res f(ak).
. ak ,
Oqa11
Y
qa2 2 Yqann
p p p p
C i j = i 6= j. f(z) C,
k, C k. = C + 1 + + n .
1
2pii
f(z)dz = 0, c,
1
2pii
C
f(z)dz =nk=1
1
2pii
k
f(z)dz =nk=1
resf(ak).
.
8.2
a - f(z). f(z) = c0 ++c1(za)+ + cn(za)n+ + c1 1za . (za)f(z) = c1 + c0(za)++c1(z a)2 + . . . . lim
za(z a)f(z) = c1 = res f(a). n. f(z) =
=k=0
ck(za)k+ c1za + + cn(za)n . (za)nf(z) = cn+cn+1(za)+ . . . . n 1 . c1(n 1)! . , z a, lim
zadn1dzn1 ((z a)nf(z)) = c1(n 1)! . , c1 =
= 1(n1)! limzadn1dzn1 ((z a)nf(z)) .
1.
1.
f(z) =sin z2
z3 pi4z2
75
-
.. z = 0
z = pi4 . z = 0
limz0
f(z) = limz0
sin z2
z2 limz0
1
z pi4= 4
pi.
, z = 0 f(z). res f(0) = 0.
z = pi4 limzpi4f(z) =, .. z = pi/4 (-
) f(z). res f(pi4 ) = limzpi4f(z)(z pi4 ) = limzpi4
sin z2
z3 pi4z2(z pi4 ) =
= limzpi4
sin z2
z2= 16pi2 sin
pi2
16 .
2.
f(z) =ez
(z + 1)3(z 2) .
. f(z) z = 1 z = 2. z = 1 f(z) .
res f(1) = 12!
limz1
d2
dz2
(ez
(z + 1)3(z 2) (z + 1)3
)=
= limz1
1
2 (z
2 6z + 10)ez(z 2)3 =
17
54e.
z = 2 - ,
res f(2) = limz2
ez
(z + 1)3(z 2) (z 2) =e2
27.
. f(z) z0
f(z) =(z)
z,
(z0) 6= 0, (z0) = 0, (z0) 6= 0, .. z0 - () f(z),
res f(z0) =(z0)
(z0).
76
-
- :
res f(z0) = limzz0
(z)
(z)(z z0) = lim
zz0(z)
(z)(z0)zz0
=(z0)
(z0).
3.
f(z) =1
z4 + 1
.. f(z) - , ..
z4 + 1 = 0. z1 = eipi4 , z2 = ei 34pi, z3 = ei 34pi, z4 = eipi4 . ,
res f(z1) =1
4z3
z=z1
=1
4ei
34pi,
res f(z2) =1
4z3
z=z2
=1
4ei
94pi,
res f(z3) =1
4z3
z=z3
=1
4ei
94pi,
res f(z4) =1
4z3
z=z4
=1
4ei
34pi.
4.
f(z) = z3 sin1
z2
.. f(z) z = 0. -
f(z), .. z = 0
f(z) = z3(
1
z2 1
3!z6+
1
5!z10 . . .
)= z 1
3!z3+
1
5!z7 . . . ,
.. . c1 = 0. res f(0) = c1 = 0.
5. z = 0
f(z) =sin 3z 3 sin z(sin z z) sin z .
. z = 0 (z) = sin 3z 3 sin z, (z) = (sin zz) sin z.
77
-
, sin z :
(z) = 3z 33z3
3!+
35z5
5! 3
(z z
3
3!+z5
5! . . .
)=
= 33 33!
z3 +35 3
5!z5 = z31(z),
1(z) = 3333! + 3535! z . . . , 1(0) = 4 6= 0.
(z) =
(z
3
3!+z5
5! . . .
)(z z
3
3!+ . . .
)=
= z4( 1
3!+z2
5! . . .
)(1 z
2
3!+ . . .
)= z41(z),
1(z) =( 13! + z
2
5! . . .)(
1 z23! + . . .), 1(0) = 16 6= 0.
f(z) =z31(z)
z41(z)=
1(z)
z1(z),
1(0) 6= 0, 1(z) 6= 0, z = 0 ,
ressin 3z 3 sin z(sin z z) sin z
z=0
= limz0
1(z)
z1z z = 1(0)
1(0)=416
= 24.
6. f(z) = e1/z
1z .. z = 1 z = 0.
z = 1 - ,
res f(1) = limz1
e1/z
1 z (z 1) = limz1e1/z
1 = e.
z = 0 f(z) .
e1/z = 1 +1
z+
1
2!z2+
1
3!z3+ . . . ,
1
1 z = 1 + z + z2 + z3 + . . . .
e1/z
1 z =(
1 +1
z+
1
2!z2+
1
3!z3+ . . .
)(1 + z + z2 + z3 + . . .
)=
=(
1 +1
2!+
1
3!+ . . .
) 1z
+ c2 1z2
+ +k=0
ckzk,
78
-
ck 6= 0, k = 2, 3, . . . .
z, z = 0 f(z). res f(0) = c1 = 1 + 12! +
13! + = e 1.
7.
f(z) = z2 sin1
z + 1
.. z = 1.
f(z) z = 1:
z2 = ((z + 1) 1)2 = (z + 1)2 2(z + 1) + 1, (1)
sin1
1 + z=
1
z + 1 1
3!(z + 1)3+
1
5!(z + 1)5 . . . . (2)
(1) (2)
f(z) = z2 sin1
z + 1=
=((z + 1)2 2(z + 1) + 1) ( 1
z + 1 1
3!(z + 1)3+
1
5!(z + 1)5 . . .
)=
=
(1 1
3!
)1
z + 1+
2
3!
1
(z + 1)2+
(1
5! 1
3!
)1
(z + 1)3+ + (2 + (z + 1)).
- (z + 1), z = 1 res f(1) = c1 = 1 13! = 56 .
z = 0:
244.zn1
sinn z, n = 1, 2, . . . . 245.
sin 2z 2z(1 cos z)2 .
246.ez 1 z
(1 cos 2z) sin z . 247.(1 ch z) sh z
(1 cos z) sin2 z .
248.zn2
shn z, n = 2, 3, . . . . 249.
z2
ch z 1 z22.
79
-
:
250.tg z
z2 pi4z. 251. z3e1/z. 252.
ch z
(z2 + 1)(z 3) .
253.ez
14 sin2 z
. 254.ez
z3(z 1) . 255.z
(z + 1)3(z 2)2 .
256.e1/z
2
1 + z4. 257. z2 sin
1
z. 258. cos
1
z+ z3.
259.sin 2z
(z + i)(z i2)2. 260.
1 cos zz3(z 3) . 261. e
z2+ 1z2 .
262.eiz
(z2 1)(z + 3) . 263.cos z
z3 pi2z2. 264.
epiz
z i.
265.z2n
(z 1)n (n>0 - ). 266. ctg2 z. 267. sin z cos 1
z.
268. ezz1 . 269.
sin 1z1 z . 270.
e1z
1 + z.
271. ez2+1z . 272. ez sin
1
z.
2.
1. |z|=4
ez 1z2 + z
dz.
. |z| < 4 f(z) = ez1z2+z - z = 0 z = 1.
|z|=4
ez 1z2 + z
dz = 2pii (res f(0) + res f(1)) .
z = 0 - f(z),
limz0
ez 1z2 + z
= 1.
80
-
res f(0) = 0. z = 1 - () f(z).,
res f(1) = limz1
ez 1z2 + z
(z + 1) = 1 e1.
|z|=4
ez 1z2 + z
dz = 2pii(1 e1).
2. |z|=2
tg zdz.
. |z| < 2 f(z) = tg z , z = pi2 z = pi2 . , ,
res f(pi
2
)=
sin z
(cos z)
z=pi2
= 1,
res f(pi
2
)=
sin z
(cos z)
z=pi2
= 1.
|z|=2
tg zdz = 2pii(2) = 4pii.
3.
|zi|= 32
e1/z2
z2+1dz.
. |z i| < 32 f(z) = e1/z2
z2+1 : z = i - z = 0 - . z = i:
res f(i) =e1/z
2
(z2 + 1)
z=i
=e1
2i.
res f(0) z = 0. -, f(z) - , , z 1z , c1 = 0 , ,res f(0) = 0.
81
-
|zi|= 32
e1/z2
z2 + 1dz = 2pii e
1
2i=pi
e.
4. |z|=2
1
z 1 sin1
zdz.
. |z| < 2 f(z) = 1z1 sin 1z z = 1 z = 0. z = 1 - ,
res f(1) =sin 1z
(z 1)z=1
= sin 1.
z = 0 f(z) :
1
z 1 sin1
z= 1
1 z sin1
z=
= (
1 + z + z2 + . . .)(1
z 1
3!z3+
1
5!z5 . . .
)=
=(
1 13!
+1
5!. . .
)1
z+c2z2
+c3z3
+ +k=0
ckzk, ck 6=0, k=2, 3, . . . .
, - z. , z = 0 -
res f(0) = c1 = (
1 13!
+1
5! . . .
)= sin 1.
:|z|=2
1
z 1 sin1
zdz = 2pii(sin 1 sin 1) = 0.
82
-
:
273.
|z|=1
z tg piz dz. 274.
zdz
(z 1)2(z 2) , : x2/3+y2/3=32/3.
275.
|z|=2
ez
z3(z + 1)dz. 276.
|zi|=3
ez2 1
z3 iz2dz.
277.
|z|=1/2
z2 sin1
zdz. 278.
|z|=3
sinpiz
z2 zdz.
279.
|z+1|=4
z
ez + 3dz. 280.
|z|=1
z2
sin2 z cos zdz.
281.
|zi|=1
ezdz
z4 + 2z2 + 1. 282.
|z|=4
eiz
(z pi)3dz.
283.
cos z2z2 4dz, :
x2
9+y2
4= 1.
284.
e2z
z3 1dz, : x2 + y2 2x = 0.
285.
sin piz
(z2 1)2dz, :x2
4+ y2 = 1.
286.
z + 1
z2 + 2z 3dz, : x2 + y2 = 16.
287.
z sin z
(z 1)5dz, :x2
3+y2
9= 1.
288.
dz
z4 + 1, : x2 + y2 = 2x. 289.
|z|=1
z2 sin1
zdz.
290.
|z|=1/3
(z + 1)e1/zdz. 291.
|z|=2/3
(sin
1
z2+ ez
2
cos z
)dz.
83
-
8.3
z = f(z). U() = {z : |z| > R} - f(z) U(). C , U().
. f(z) - 12pii
C
f(z)dz, -
C . U() f(z)
f(z) = c0 +c1z
+c2z2
+ + c1z + c2z2 + . . . .
C, . ,
C
cnzndz = 0, n 6= 1;
C
c1dz
z= c12pii,
1
2pii
C
f(z)dz = c1.
, res f() = c1.
. , - , . . , 1z , 1.
. f(z) - , - ( ) .
. , . -, 12pii
C
f(z)dz
, C. , 12pii
C
f(z)dz f(z)
. , -
84
-
1
2pii
C
f(z)dz +1
2pii
C
f(z)dz = 0.
1. f(z) = z+1z .
. limz
z+1z = 1. z =
. f(z) = 1 + 1z . c1 = 1 ,, res f() = 1.
2. |z|=2
dz
1 + z4.
. zk (k = 1, 2, 3, 4) 1 + z4 = 0 () f(z) = 11+z4 . , |z| = 2. f(z) = 11+z4
f(z) =1
1 + z4=
1
z4 1
1 + 1z4=
1
z4 1z8
+1
z12 . . . ,
res f() = c1 = 0. 8.3
4k=1
res f(zk) + res f() = 0.
|z|=2
dz
1 + z4= 2pii
4k=1
res f(zk) = 2pii res f() = 0.
3. |z|=3
z17
(z2 + 2)3(z3 + 3)4dz.
85
-
. f(z) = z17
(z2+2)3(z3+3)4 |z| =3 zk, . 8.3,
5k=1
res f(zk) + res f() = 0.
,|z|=3
z17
(z2 + 2)3(z3 + 3)4dz = res f().
res f() :
f(z) =z17
(z2+2)3(z3+3)4=
z17
z6(1+ 2z2
)2z12(1+ 3z3
)4 = 1z 1(1+ 2z2)3 (1+ 3z3)4 . ,
1z ., res f() = 1. ,
|z|=3
z17
(z2 + 2)3(z3 + 3)4dz = 2pii.
:
292. f(z) =z3 z2 + z + 6
z2. 293. f(z) =
z + 1
z4.
294. f(z) =ez
z2. 295. f(z) = cos
1
z.
296. f(z) = e1/z2
. 297. f(z) = z3e1/z. , -
86
-
:
298.
|z|=1
z2 + 1
z3dz. 299.
|z|=2
dz
1 + z12.
300.
|z|=2
1000z + 2
1 + z1224dz. 301.
|z|=3
ez
z 1dz.
302.
|z|=1
z2 sin1
zdz. 303.
|z|=3
z9
z10 1dz.
8.4
.
1
2pii
f (z)f(z)
dz
f(z) - ( , f
(z)f(z)
f(z) : ddz(ln f(z))).. - - . f(z)
, , , -, . , f(z) . a1, . . . , an f(z) , 1, . . . , n , b1, . . . , bm 1 . . . , m f(z) , . f(z) , ..
1
2pii
f (z)f(z)
dz =ni=1
i mj=1
j.
. . - F (z) = f
(z)f(z) . F (z) ,
f(z). ai f(z). - ai :
f(z) = Ai(zai)i + . . . , f (z) = Aii(zai)i1 + . . . , Ai 6=0.
F (z) =Aii(z ai)i1 + . . .Ai(z ai)i + . . . =
Aii + . . .
Ai(z ai) . . . =Aii + . . .
(z ai)(Ai + . . . ) .
87
-
, F (z) ai . F (z) ai resF (ai) = AiiAi = i.
bj - f(z).
f(z) = Bj(z bj)j + . . . , f (z) = Bji(z bj)j1 + . . . , Bj 6= 0. F (z) :
F (z) =Bjj(z bj)j1 + . . .
Bj(z bj)j + . . . =Bjj + . . .
Bj(z bj) + . . . .
, F (z) bj . F (z) bj resF (bj) =
BjjBj
= j. , F (z) -
ni=1
i mj=1
j.
ni=1
i f(z)
, mj=1
j - ,
, . .
8.5
1 ( ). ( -) n : f(z) = a0zn + a1zn1 + ++an1z + an, n 1, n , , .
. f(z) z =, - n. lim
z f(z) =, R , z, - |z| R |f(z)| > 1. .. f(z) |z| < R. N - ( , ). , - , :
N =1
2pii
f (z)f(z)
dz, = {z : |z| = R}.
, N = n. , 1
2pii
f (z)f(z)
dz = N
88
-
f(z)f(z) ,
, N .
f(z) = zna0 + zn1a1 + + an = zn(a0 + a1
z+ + an
zn) = zn(z).
(z) = a1z2 2a2z3 nanzn1 ,
(z)(z)
= a1z2 +
2a2z3 + + nanzn+1
a0 +a1z + + anzn
= a1z2 +
2a2z3 + + nanzn+1
a0(1 +a1a0
1z + + ana0 1zn )
.
: 11+z = 1 z + z2 z3 + z4 . . . . (z)(z)
= 1a0
(a1z2
+2a2z3
+ . . .+nanzn+1
)(1a1
a01za2a0 1z2 . . .an
a0 1zn
+ . . .
)=
= a1a0 1z2
+b1z3
+b2z4
+ . . . , bk C.
f(z)f(z) :
f (z)f(z)
= (ln z)z = (ln zn(z))z = (n ln z + ln(z))
z =
=n
z+(z)(z)
=n
z a1a0 1z2
+ res f(z)f(z)
z=
= n.
, N = n. 1 . 2 ( ). f(z) -
, , - a1, . . . , an, - . - f(z) 2.
f(x)dx = 2piink=1
res f(ak).
. f(z) :
f(z) =c2z2
+c3z3
+ . . . .
89
-
6-
RR -
=
q a1q a2
q anq q q
R, . -, :
+[R,R]
f(z)dz =
RR
f(x)dx+
f(z)dz =
= 2piink=1
res f(ak).
, limR
f(z)dz = 0. :
|f(z)| |c2|R2
+|c3|R3
+ = 1R2
(|c2|+ |c3|
R+ . . .
).
R, :
|c3|R
+|c4|R2
+ < 1. |f(z)| < |c2|+ 1R2
.
,
f(z)dz.
f(z)dz
|f(z)|ds |c2|+ 1R2
ds =|c2|+ 1R2
piR =|c2|+ 1
Rpi.
limR
f(z)dz = 0.
limR
RR
f(x)dx+
f(z)dz
=
f(x)dx = 2piink=1
res f(ak).
.
+0
x2
(x2 + a2)2dx, a > 0.
90
-
. f(x) = x2
(x2+a2)2 - , -:
+0
x2
(x2 + a2)2dx =
1
2
+
x2
(x2 + a2)2dx.
f(z) = z2
(z2+a2)2 . - z = ai.
res f(ai) = limzai
d
dz(f(z)(z ai)2) =
= limzai
d
dz
(z2
(z + ai)2
)= lim
zai2aiz
(z + ai)2=
1
4ai.
2 8.5,
+0
x2
(x2 + a2)2dx =
1
2
+
x2
(x2 + a2)2dx =
1
2 2pii 1
4ai=
pi
4a.
:
304.
+0
x2 + 1
x4 + 1dx. 305.
+
dx
(x2 + a2)(x2 + b2), a > 0, b > 0.
306.
+
dx
(x2 + 1)3. 307.
+
dx
(1 + x2)n+1.
308.
+
x dx
(x2 + 4x+ 13)2. 309.
+
dx
(x2 + a2)2(x2 + b2)2.
310.
+0
x4 + 1
x6 + 1dx. 311.
+
x2m
1 + x2ndx.
312.
+
dx
1 + x6. 313.
+
dx
(x2 + 2x+ 2)2.
91
-
314.
+
x4dx
(a+ bx2)4, a > 0, b > 0.
315. +
dx
(1 + x2)=
1 3 . . . (2n 1)2 4 6 . . . 2n pi.
92
-
1. i. 2. 1. 3. 1r2+i2r sin1+r22r cos . 4. 32 i. 5. 46. 6. cos 2 + i sin 2.7. a
2b2a2+b2+i
2aba2+b2 . 8. 4. 9. x =
2017 , y = 3617 . 10. x = ba2+b2 , y = aa2+b2 .
11. . 12. x= 411 , y= 511 . 13. x=1+i, y=i.14. x = 2 + i, y = 2 i. 15. x = 3 11i, y = 3 9i, z = 1 7i.16. 2(a
2b2)(a2+b2)2 . 18. z1 = 0, z2 = 1, z3 = 12 + i
3
2 , z4 = 12 i
32 .
19. |z| = 5, arg z = arctg 34 . 20. |z| = 4, arg z = 23pi.21. |z| = 52, arg z = arctg 17 pi. 22. |z| = 1, arg z = 45pi.23. |z| = 5, arg z = arctg 34 . 24. |z| = 1, arg z = 2pi .25. 2(cospi + i sinpi). 26. 2(cos pi2 + i sin
pi2 ). 27. 2(cos
34pi + i sin
34pi).
28.
2(1 sin) (cos (pi4 + 2)+ i sin (pi4 + 2)) . 29. 1(cos + i sin).30. 219(1 + i3). 31. 210(1 + i). 32. 1728. 33. 1. 36. 1i
2.
37. 12(1 + i). 38. 12(
3 + i), i.
39. (cos pi8 i sin pi8 ), (cos 38pi + i sin 38pi). 40. 1, i.41.
3
42 (1 + i),
6
2( cos pi12 + i sin pi12), 6
12(sin pi12 i cos pi12).42. (3 i).43. 10
2(cos 6 + i sin 6), 10
2(cos 78 + i sin 78), 10
2(cos 150 + i sin 150),10
2(cos 222 + i sin 222), 10
2(cos 294 + i sin 294).44. , 2 .45. r = 1 , ( ).46. , r = 1/2 .47. r = 8 z = 5i.48. r = 4 z = 1 + i.49. , arg z = pi4 arg z =
pi2
r = 1 r = 2 z = i.50. , OY .51. y = 0 y = 1, .52. , r1 = 1 r2 = 2 z = (2 + i). .53. , y = x.54. x = 1 x = 2.55. u = x+ 2xy, v = y2 x2 y.56. u = x2 y2, v = 1 + 2xy.57. u = 3xy2 x3, v = 1 3x2y + y3.58. xx2+y2 , v =
yx2+y2 .
93
-
59. u = x2xyy+1(x+1)2+y2 , v =x2+yy2+x(x+1)2+y2 .
60. u = x2y2x2+y2 , v = 2xyx2+y2
61. u = 2x 1, v = 2y.62. u = x2 y2 + x, v = (2x+ 1)y.63. u = xx2+y2 , v = yx2+y2 .64. u = ex cos y, v = ex sin y.65. u = ex
2y2 cos 2xy, v = ex2y2 sin 2xy.66. u = sinx ch y, v = cosx sh y.67. u = chx cos (y 1), v = shx sin (y 1).68. u = e(x
2y2) ln 24kpixy cos (2kpi(x2 y2) + 2 ln 2 xy),v = e(x
2y2) ln 24kpixy sin (2kpi(x2 y2) + 2 ln 2 xy), k Z.69. u = chx cos y, v = chx sin y.70. u = sinx cosx
ch2 ysin2 x , v =sh y ch y
ch2 ysin2 x .71. ) = 34 , 0 = pi2 ; ) = 54 , 0 = pi.72. = ch 1, 0 =
pi2 .
73. = pi, 0 = pi2 .74. = cos2 (ln 3), 0 = 0.75. ) = 0, - ;
) = e2kpi, = ln 10 + 2mpi, k,m Z;) = 9e2kpi, = ln 3 + 2mpi, k,m Z.
76. ) (2k 12
)pii, k Z; ) 2k + 12pii, k Z;
) ln
2 + (2k 34
)pii, k Z; ) ln
13 + (2kpi arctg 23
)i, k Z;
) (2k+12
)pi + 2mpii, k,m Z.77. ) e(2k+
12 )pi, k Z; ) e2kpi, k Z;
) e
2(2k+1)pii, k Z; ) e(4k+ 12 )pi, k Z;) e(i1)(2k+
16 )pi, k Z; ) 2 32e3(2kpipi4 )3(pi4 +ln
22kpi)i, k Z.
78. 1. 79. 0. 80. . 81. 0. 82. 13 . 83. i. 84. i.85. . 86. 0. 89. i. 90.
2. 91. i. 92. 2i.
98. ) ; ) ; ) ; ) ; ) ; ) ; ) ; ) .99. ) ; ) ; ) ; ) . 102. pi2 . 103. 14(e2 1)(1 + i).104. ) 2pii; ) 2pii. 105. 0. 106. 0. 107. (i 1)ei. 108. ) 2 + i; ) 6 + 2i.109. 2(1 + i). 110. 1. 111. 35(i 1). 112. ) e cos 1 1 + ie sin 1;) e cos 1 1 + ie sin 1. 113. (1 + i sh 1). 114. 7e2 + (3 2i)ei.115. e1 1. 116. cos 1 sin 1 ie1. 117. 1 cos 1 + i(sin 1 1).118. 18(pi
2
4 + 3 ln2 2) + ipi8 ln 2. 119. pi
2
8 . 120.14(1 cos (2 + 2i)).
121. (tg 1 + 12 tg2 1 + 12 th2 1) + i th 1. 122. (14 sh 2 + 12)i. 123. 43 .
94
-
124. 0. 125. ln
sh2 1 + cos2 1 + i arctg (tg 1 th 1).126. ) i; ) 2i; ) 2i. 127. ) e(2 ei 1); ) 1 + ei(e 2).128. pii. 129. pie1. 130.
pi
2i. 131. pi sh 1. 132. 0. 133. 23pi chpi i.
134. 0. 135. pi45i. 136. pi. 137. 0. 138. pii. 139. 2pii.140. pi(pi+2)
2
8 i. 141. 0. 142. pii27 . 143. pi2
2 sh 1. 144. pi3i.
145. 0. 146. 2pii. 147. 1+i2 eipi. 148. .149. . 150. . 151. .152. . 153. . 154. .155. . 156. . 157. .158. R = 1. 159. R = 1. 160. R =
2. 161. R =.
162. R = 1. 163. R =. 164. R = 1. 165. R = 1.166. R = 1. 167. R =. 168. R = 1. 169. R = e1.170. sin 1 + 2(z + 1) cos 1 + 222! (z + 1)2 sin 1 2
3
3! (z + 1)3 cos 1 . . . ,
R =.171. 1
2
(1 + (z + pi4 ) 12!(z + pi4 )2 13!(z + pi4 )3 + . . .
), R =.
172.e(1 + 12(2z 1) + 12!22 (2z 1)2 + 13!23 (2z 1)3 + . . .
), R =.
173. 15(
1 + 35(z + 2) +32
52 (z + 2)2 + 3
3
53 (z + 2)3 + . . .
), R = 53 .
174. 15 925z 41125z2 . . . , R = 1.175. iz + z3 + iz5 z7 + . . . , R = 1.176. 1 + 12
(z2
2! +z4
4! +z6
6! + . . .), R =.
177. 12
(z2
2! +z4
4! +z6
6! + . . .), R =.
178. ln 2 12(z2 +
z2
24 +z3
38 + . . .), R = 2.
179. ln 2 + z2 5z2
24 +7z3
38 . . . , R = 1.180. 12 122z + 13!23z3 + 35!25z5 + . . . , R = pi.181. 12 122z + 12!22z2 13!23z3 + . . . , R =
ln2 (23) + pi2.
182. 16 +162z 42!63z2 + 13!63z3 + . . . , R =
ln2 5 + pi2.
183. ln 2 12z + 12!22z2 14!23z3 + . . . , R = pi.184. 12!z2 24!z4 166! z6 + . . . , R = pi2 .185. ln 2 z24 z
4
44! z6
26! + . . . , R = pi.186. e(1 + z + 32!z
2 + 133! z3 + . . . ), R = 1.
187. ) z = 0 - , z1,2 = 2i - ;) zn = pi (n Z) - .
188. ) z = 0 - , zn = pin (n Z) - ;) z = 0 - , zn = npii (n Z) - .
95
-
189. ) zn = (2n+ 1)pii (n Z) - ;) zn = (4n+ 1)
pi
2i (n Z) - .
190. ) z = pii - , zn = npii (n Z) - ;
) zn = 3
(2n+ 1)pi
2, zn =
(2n+ 1)
pi
2 1 i
3
2(n Z) - .
191. ) z1,2 = pii - , zn = (2n+ 1)pii (n Z) - ;) .
192. . 193. .194. . 195. .196. . 197. .198. . 199. .
200. 1z z3! + z3
5! z5
7!+ . . . .
201. 92!z 84!z3 + 326! z5 . . . .202. 1z + 1 +
z2! +
z2
3! + . . . .
203. 1z3 +1z2 +
12! 1z + 13! + z4! + . . . .
204. z3 + z2 + z2! +13! +
14! 1z + . . . .
205. z4 z22! + 14! 16! 1z2 + . . . .206. 4
2
2! 12z3 44
4! 12z5 + 46
6! 12z7 . . . .207. 12! z
2
4! +z4
6! z6
8! + . . . .
208. 1 + z2! +z2
3! +z3
4! + . . . .
209. 2z4 12! 1z2 + 14! z2
6! + . . . .
210. 1z2 12! 1z + 13! z4! + . . . .211. 1z+1 1(z+1)2 .212. sin 2z2 sin 22! (z 2) cos 23! (z 2)2 + . . . .213. (1 i) + (z + i) + ( 12! 11!) 1z+i + ( 13! 12!) 1(z+i)2 + . . . .214. )
n=1
2n1zn 13
n=0
(z3
)n; )
n=1
3n12n1zn .
215. ) 1z n=0
(1)nz2n; )n=0
(1)nzn+2 .
216. ) ; ) 15(
22+1z3 2
3+2z4 +
241z5 2
52z6 + . . .
).
217.n=1
(1)n1zn +
n=0
(1)n2n+1 z
n.
218. )n=1
[n (1)n2n
]zn1; )
n=1
nzn+1 +
n=0
(1)n2n+1 z
n;
) 1z +n=1
n+(2)nzn+1 .
96
-
219.n=1
1(z+2)n
n=0
(z+2)n
3n+1 .
220. .
221. 1z1 + 5n=2
2n2(z1)n .
222. z +n=0
(n+2)4n+1z2n+1 .
223. 112n=0
z2n+1
4n 13n=1
1z2n1 .
224. i2(zi) + 14n=0
(1)n(2i)n (z i)n.
225.n=1
n4n1(z+2)n+3 .
226. ) ; ) ;) .
227. ) ; ) .228. ) zn = (4n+ 1)
pi
2, n Z - ;
) z = 0 - .229. ) z = 2 - ;
) z = 0 - .230. ) z = 0 - , z = 1 - ;
) z = 0 - , z = 2npii (n = 1,2, . . . ) - .
231. ) z = 0 - ;) z = 1 - ;) z = 0 - .
232. ) z = 0 - , z = 2pin (n = 1,2, . . . ) - ;
) z =pi
2+ 2pin (n Z) - ,
z = pi2
+ 2pin (n Z) - ;) z = pi - , z = pin (n = 0,1,2,3, . . . ) - .
233. . 234. .235. . 236. .237. .238. z = 0 - , z = 1 - .
97
-
239. . 240. .241. . 242. .243. .244. 1. 245. 16/3. 246. 1. 247. 1. 248. 0. 249. 0.250. res f(0) = 0, res f(pi4 ) =
4pi , res f(
pi2 +pin) =
8pi2(2n+1)(4n+1) , n Z.
251. res f(0) = 1/24.252. res f(i) = 1+3i20 cos 1, res f(1) = 13i20 cos 1, res f(3) = ch 310 .253.
res f((1)npi6 + pin
)=
2
3epi6 +2pin,
23e
pi6 +pi(2n1),
res f((1)n+1pi6 +pin
)=
23e
pi6 +2pin,
23epi6 +pi(2n1).
254. res f(0) = 52 , res f(1) = e.255. res f(1) = 127 , res f(2) = 127 .256. res f(0) = 0, res f(z1) = 1+i42ei, res f(z2) =
(1i)ei4
2,
res f(z3) =(1+i)ei
4
2, res f(z4) = (1+i)e
i
4
2,
zk (k = 1, 2, 3, 4) - z4 + 1 = 0.257. res f(0) = 16 . 258. res f(0) = 0.259. res f(i) = 49 sh 2 i, res f
(i2
)= 49(e+ 2e1)i.
260. res f(0) = 16 , res f(3) = 227 sin2(
32
).
261. res f(0) = 0.
262. res f(3) = 18e3i, res f(1) = ei4 , res f(1) =
ei
8 .
263. res f(0) = 4pi2 , res f(pi2 ) = 0.264. res f(i) = 1.265. res f(1) = 2n(2n1)(2n2)...(2n(n2))(n1)! .266. res f(npi) = 0, n Z.267.
n=1
1(2n1)!(2n)! z = 0.
268. e z = 1.269. sin 1 z = 0; sin 1 z = 1.270. 1 e1 z = 0; e1 z = 1.271. e1 1 z = 0.272.
n=0
(1)n(2n)!(2n+1)! z = 0.
273. 0. 274. 0. 275. (1 2e1)pii. 276. 2(1 e1)pii. 277. 13pii.278. 0. 279. 43 ln 3 pii. 280. 2pii.
98
-
281. (cos 1 + sin 1 + i(sin 1 cos 1))pi2 . 282. pii. 283. 0. 284. 2piie2
3 .
285. pi2i. 286. 2pii. 287. sin 14 cos 112 pii. 288. pii2 . 289. 0.290. 3pii. 291. 0.292. z = - .293. z = - .294. z = - .295. z = - .296. z = - .297. z = - .298. 2pii. 299. 0. 300. 0. 301. 2piei. 302. pi3 i. 303. 2pii.304. pi
2. 305. piab(a+b) . 306.
38pi. 307.
(2n)!(n!)2 2
2npi. 308. pi27 .309. pi2(b2a2)3
(5b2a2b3 +
b25a2a3
). 310. 23pi. 311.
pin sin 2m+1n pi
. 312. 23pi.
313. pi2 . 314.pi
16a3/2b5/2.
99
-
1. .. -
. - .: , 1977.2. .. . -
.: , 1978.3. .., .. -
. - .: , 1973.4. .., .., ..
. - .: , 1976.5. .., .., .. -
. . . - .: -, 1981.
100
-
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 31. . . . . . . . . . . . . . . . . . . . . 4
1.1 . . . . . . . . . . 41.2 . . . . . . . . . . . 51.3 . . . . 51.4
. . . . . . . . . . . . . . . . . . . . 61.5 . . . . . . . . . . . . . . 61.6 . . . . . . . . . . . 71.7 . . . . . . . . . . . . . . . . . . . . . 7
2. . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1
. . . . . . . . . . . . . . . . . . . . . . . . . 123.
. . . . . . . . . . . . . . . 16
4. . - . . . . . . . . . . . . . . . . . . 204.1
. . . . . . . . . . . . . . . . . . . . . . . . . 235.
. . . . . . . . . . . . . . 255.1 . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2 . . . . . . . . . . . . . . . . . . . . . . . 285.3
. . . . . . . . . . . . . . . . . . . 335.4 . . . . . . . . . . . . . . . 395.5 . . . . . . . . . 435.6 . . . . . . . . . . . . 44
101
-
6. . . . . . . . . . . . . . . . . . . . . . . . 486.1 . . . . . . 486.2 -
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506.3 , 51
7. . . . . . . . . . . . . . . . . . . . . . . . . . . 587.1
. . . . . . . . . . . . . . . . . . . . . . . . 587.2 . . . . . . . . 607.3 . . . . . . . 617.4
. . . . . . . . . . . . . . . . . . . 677.5
. . . . . . . . . . . . . . . . . . . . . . . 698. . . . . . . . . . . . . . . . . . . . . . . . 74
8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
8.2 . . . . . . . 758.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 848.4 . . . . . . . . . . . . 878.5 . . . . . . . . . . . . . . . 88
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
102
. . -
. . . .
,