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b
(
x
)
B
0
ik
y
k
A
−
ξ
y
(
x
)=
.
k
2
||
Substituting this expression into (4.44) results in
=
k
A
−
ξ
x
(
x
)
,
b
||
(
x
)
B
0
d
dx
k
2
dx
ξ
x
(
x
)=
b
(
x
)
B
0
k
y
k
A
−
d
−
1+
.
(4.57)
k
2
||
Let the condition of FLR (4.56) be fulfilled at
x
=
x
1
, i.e. function
k
A
(
x
)
k
2
has a simple zero at
x
=
x
1
. Then the point
x
1
is a regular
singular point of (4.57). Rewrite (4.57) in the canonical form. For this multi-
ply each equation of the system by
x
−
−
x
1
. The obtained equations in matrix
form are
x
1
)
d
U
dx
(
x
−
=
T
(
x
)
U
(
x
)
,
(4.58)
where
x
1
)
k
A
(
x
)
⎛
⎞
k
2
0
(
x
−
−
⎝
⎠
T
(
x
)=
x
1
)+
k
y
(
x
,
−
x
1
)
−
(
x
−
0
k
A
(
x
)
−
k
2
U
=
b
/
B
0
ξ
x
.
If Alfven velocity changes smoothly near the resonance point, then
T
(
x
)can
be expanded into a power series
x
1
)
2
+
T
(
x
)=
T
0
+
T
1
(
x
−
x
1
)+
T
2
(
x
−
···
,
T
1
=
00
−
,
T
n
=
0
s
n
q
n
0
,
n
=0
,
2
,
3
,...,
1+
q
1
0
where
k
y
k
A
(
x
1
)
k
A
(
x
1
)
k
y
q
0
=
,
1
=
−
2(
k
A
(
x
1
)
)
2
,
k
A
(
x
1
)
2
4(
k
A
(
x
1
)
)
3
−
,
k
A
(
x
1
)
6(
k
A
(
x
1
)
)
2
q
2
=
k
y
x
=
x
1
dk
A
(
x
1
)
dx
2
=
k
A
(
x
1
)
,k
A
(
x
1
)
=
s
0
=0
,
1
=0
,
.
The fundamental matrix solution of (4.58) can be presented as (e.g. [20])
U
(
x
)=
P
(
x
)exp(
T
0
ln(
x
−
x
1
))
,
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