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,
.
.
( )
,xaxaxxy
m m
m
m
m
m
∑∑
∞
=
∞
=
+
==
0 0
σσ
(16)
−,...,...,,,
10 m
aaaσ
, ,
( )
xy
(7).
(16)
( )
xy
′
( )
xy
′
′
, (7):
( )( ) ( )
0
1
0
2
0
2
00
=−+
+++−++
∑∑
∑∑
∞
=
+
∞
=
++
∞
=
+
∞
=
+
m
m
m
m
m
m
m
m
m
m
m
m
xaxa
xamxamm
σσ
σσ
ν
σσσ
:
( )
( )
( )
( )
[ ]
{ }
∑
∞
=
+
−
+
=+−++
+−++−
0
2
2
2
1
1
2
2
0
22
.0
1
m
m
mm
xaam
xaxa
σ
σσ
νσ
νσνσ
,...3,2,,,
1
=
++
mxxx
mσσσ
:
( )
,0
0
22
=− aνσ
(17)
( )
( )
,01
1
2
2
=−+ aνσ
(18)
( )
(
)
...,3,2,0
2
2
2
==+−+
−
maam
mm
νσ
(19)
(7),
,0
0
≠a
ν
σ ±=
.
ν
σ
=
.
(18) ,
,0
1
=a
(19) :
( )
...,3,2,
2
2
=
+
=
−
m
mm
a
a
m
m
ν
(20)
...,2,1,12
=
+
=
kkm
,
,0
1
=a
(20) ,
:
...,2,1,0
12
==
+
ka
k
(21)
,
.
.
∞ ∞
y( x ) = x σ
∑a m x = ∑ am x m+σ ,
m
(16)
m =0 m=0
σ , a0 , a1 ,..., am ,... − , ,
y( x ) (7).
(16) y′(x) y′′(x ) , (7):
∞ ∞
∑ (m + σ )(m + σ − 1)a
m=0
m x m+σ
+∑ (m + σ ) am x m+σ +
m=0
∞ ∞
+ ∑ am x m+σ +2
− ∑ν 2 am x m+σ = 0
m=0 m=0
:
(σ 2
)
−ν 2 a0 xσ + (σ + 1) −ν 2 a1 xσ +1 +( 2
)
{[ ] }x
∞
+ ∑ (m + σ ) −ν 2 am + am−2 m+σ
= 0.
2
m =0
xσ , xσ +1 , xσ +m , m = 2,3,...
:
(σ 2
)
−ν 2 a0 = 0 , (17)
((σ + 1) −ν ) a = 0 ,
2 2
(18)
((m + σ ) −ν ) a + a
1
= 0 , m = 2, 3, ...
2 2
m m−2 (19)
(7), a0 ≠ 0 ,
σ = ±ν . σ =ν .
(18) , a1 = 0 , (19) :
am − 2
am = , m = 2, 3, ...
m(m + 2ν )
(20)
m = 2k + 1, k = 1, 2, ... , a1 = 0 , (20) ,
:
a2 k +1 = 0 , k = 1, 2, ... (21)
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