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Hence, we obtain for
P
1
,
P
2
the considerably simplified expressions
P
1
=
T
1
+[
T
0
T
1
]
,
P
2
=
1
2
T
1
T
0
T
1
−
T
1
T
0
+
C
2
+
1
1
4
T
0
T
2
T
0
,
4
[
T
0
C
2
]
−
C
2
=
T
2
+
T
1
.
The coecients in the singular part of (6.17) are even more simple:
P
0
T
0
=
T
0
,
P
1
T
0
=
T
1
T
0
,
P
2
T
0
=
1
2
(
T
2
+
T
1
)
T
0
+
1
4
T
0
T
2
T
0
.
Finally, we obtain in block form
P
1
T
0
=
0
,
P
1
=
R
0
,
0
S
0
R
0
Q
0
0
Q
1
+[
Q
0
R
0
]
R
0
⎛
⎞
(
S
1
+
S
0
R
0
)
Q
0
0
R
1
+
R
0
+
1
2
Q
0
S
1
Q
0
P
2
T
0
=
1
2
⎝
⎠
.
R
0
Now we obtain the exact expressions describing plasma behavior near a
resonance point
x
0
.
It is convenient to present the matrix solution (6.5) in a
block form
U
(1)
U
(4)
=
P
(3)
Q
0
ln
x
+
P
(1)
P
(4)
,
U
(3)
P
(3)
0
U
≡
U
(2)
P
(4)
Q
0
0
P
(2)
where
x
=
k
y
(
x
x
0
). It is evident from this expression that the singularity
ln
x
exists in many solutions of the fundamental system (namely in half of
the columns of matrix
U
(
x
)). However, it turns out that it is possible to
find a fundamental matrix
U
(
x
) in which only one column has a logarithmic
singularity, but all other columns have no singularity at
x
=
x
0
. Due to a
special form of matrix
Q
, all the columns of block
P
(
i
)
Q
0
(
i
=3
,
4) in the
matrix of coecients at ln
x
are collinear to the
s
-th column of block
P
(
i
)
:
−
(
P
(
i
)
Q
0
)
nm
=
P
(
i
)
nν
δ
νs
q
m
=
q
m
P
(
i
)
ns
.
Hence, subtracting from all other columns of the left-hand part of matrix
U
,
the
s
-th column of
U
multiplied by
q
m
/q
s
(
q
s
= 0), see (6.14)) we eliminate
a singularities from all columns of
U
except of the
s
-th column. Then, we get
q
m
q
m
P
(1)
ns
ln
x
+
P
(
i−
2)
q
s
(
q
s
P
(
i
)
ns
ln
x
+
P
(
i−
2)
−
)
nm
ns
q
m
=
P
(
i−
2)
nm
q
s
P
(
i−
2)
−
,
i
=3
,
4
,m
=
s.
(6.19)
ns
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