Geoscience Reference
In-Depth Information
where
x
1
)
2
P
2
+
···
is a regular matrix function of
x
and
1
is a unit matrix. By virtue of equality,
P
(
x
)=
1
+(
x
−
x
1
)
P
1
+(
x
−
T
0
=
0
(
0
is a zero matrix) in the expansion of the exponent into a power series all
the powers above the first turn into zero and we obtain
U
(
x
)=
1
+(
x
···
x
1
)
2
P
2
+
−
x
1
)
P
1
+(
x
−
×
(
1
+
T
0
ln(
x
−
x
1
))
.
(4.59)
Calculations shown in Chapter 6 yield
=
D
1
k
A
(
x
1
)
R
1
(
x
)+
D
2
S
1
(
x
)+
R
1
,
b
(
x
)
B
0
(4.60a)
S
2
(
x
)+
R
2
k
A
(
x
1
)
,
ξ
x
(
x
)=
D
1
R
2
(
x
)+
D
2
(4.60b)
1
x
x
1
+
S
3
(
x
)+
R
3
.
ik
y
k
A
(
x
1
)
ξ
y
(
x
)=
D
1
ik
y
R
3
(
x
)+
D
2
(4.60c)
−
Here we denote
R
i
=
k
y
R
i
(
x
)ln[
k
y
(
x
−
x
1
)]
,i
=1
,
2
,
3. Functions
R
n
(
x
)and
S
n
(
x
)havetheform
x
1
)
2
x
1
)
2
R
1
(
x
)=
(
x
−
(
x
−
R
2
(
x
)=1+
k
y
+
···
,
+
···
,
2
4
x
1
)
2
4
R
3
=
x
−
x
1
(
x
−
k
y
+
···
,
S
1
(
x
)=1
−
+
···
,
2
1+
(
x
k
y
k
A
(
x
1
)
2(
k
A
(
x
1
)
)
2
k
A
(
x
1
)
2
k
A
(
x
1
)
S
2
(
x
)=
−
−
x
1
)+
···
,S
3
(
x
)=
−
+
···
.
The coecients
D
1
and
D
2
are arbitrary constants which can be found
from the boundary conditions. Relations connecting electric
E
and magnetic
b
fields with displacement
ξ
reduce for a plane wave
exp (
iωt
−
i
kr
)to
(see 4.32)
E
x
B
0
=
i
ω
E
y
B
0
i
ω
c
ξ
y
,
=
−
c
ξ
x
,
b
x
B
0
b
y
B
0
b
B
0
=
ik
ξ
x
,
=
−
ik
ξ
y
,
=
−
∇
·
ξ
.
⊥
The main singularity of the field near the resonance point
x
1
can be found
without determining the coecients
D
1
and
D
2
. Note that at
x
x
1
pertur-
bations of the longitudinal magnetic field
b
tend to a constant
b
0
and have a
logarithmic singularity
→
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