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( )( )
rDBAu ln
00000
++=ϕ
.
ϕ
,
0
0
=
B
,
,
0
0
=
D
.
2
0
0
A
u =
.
(74) ,
....3,2,1,0
=
=
kD
k
, (68)
( ) ( )
∑
∞
=
++=
1
0
sincos
2
;
n
n
nn
rnBnA
A
ru ϕϕϕ
(75)
(75) (67), :
( ) ( )
∑
∞
=
++=
1
0
0
sincos
2
n
n
nn
rnBnA
A
f ϕϕϕ
. (76)
,
n
A
n
B
( ) ( )
,....2,1,0,sin
1
,cos
1
00
===
∫∫
−−
nntdttf
r
Bntdttf
r
A
n
n
n
n
π
π
π
π
ππ
(75)
n
A
n
B
(76), :
( )
( ) () ( )
=
−+=
∑
∫∫
∞
=
−−
n
n
r
r
dttntfdttfru
0
1
cos
1
2
1
; ϕ
ππ
ϕ
π
π
π
π
( ) ( )
.cos21
2
1
1
0
dttn
r
r
tf
n
n
−
+=
∑
∫
∞
=
−
ϕ
π
π
π
(77)
0
rr <
( )
( ) ( )
[ ]
( )
( )
( )
( )
( )
=
+−−
−
=
−
+
−
+=
=+
+=
−
+
−−
−−
−
−
−−−
∞
=
∞
=
∑∑
2
00
2
0
0
0
0
0
1
0
1
0
cos21
1
11
1
1cos21
r
r
t
r
r
r
r
e
r
r
e
r
r
e
r
r
e
r
r
ee
r
r
tn
r
r
ti
ti
ti
ti
tintin
n
n
n
n
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕϕ
( )
.
cos2
2
0
2
0
22
0
rtrrr
rr
+−−
−
=
ϕ
(78)
u0 = ( A0 + B0ϕ )( 0 + D0 ln r ) .
ϕ, B0 = 0 ,
, D0 = 0 .
A0
u0 = .
2
(74) ,
Dk = 0, k = 1,2,3....
, (68)
∞
u (r; ϕ ) = + ∑ ( An cos nϕ + Bn sin nϕ ) r n
A0
(75)
2 n=1
(75) (67), :
∞
f (ϕ ) = + ∑ ( An cos nϕ + Bn sin nϕ ) r0n .
A0
(76)
2 n=1
, An Bn
π π
∫−π f (t ) cos ntdt, Bn = πr0n ∫ f (t ) sin ntdt, n = 0,1,2,....
1 1
An = n
πr0 −π
(75) An Bn (76), :
π π n
1 ∞ r
u(r; ϕ ) = ( ) ( ) ( )
1
2π ∫−π f t dt + ∑
π n=1 −∫π
f t cos n t − ϕ dt =
r0
π ∞
r
n
f (t ) 1 + 2∑ cos n(t − ϕ ) dt .
1
=
2π ∫
−π n =1 r0
(77)
r < r0
r
n
n
r in (t −ϕ )
[ ]
∞ ∞
1 + 2∑ cos n(t − ϕ ) = 1 + ∑ e + e −in (t −ϕ ) =
n =1 r0 n =1 r0
2
r i (t −ϕ ) r −i (t −ϕ ) r
e e 1 −
=1+ 0
r r
+ 0 = r0 =
2
r i (t −ϕ ) r −i (t −ϕ )
1− e 1− e 1 − 2 cos(t − ϕ ) +
r r
r0 r0 r0 r0
r02 − r 2
= 2 .
r0 − 2r0 r cos(t − ϕ ) + r 2
(78)
