Geoscience Reference
In-Depth Information
8
<
p
1
k
þ
1
x
2k
1
ð
2k
1
Þð
k
1
Þ!
2
k¼1
ð
1
Þ
when
j
x
j
x
1
;
Þ
1
k¼0
p
exp
ð
x
2
2
2
k
x
2k
þ
1
ð
2k
þ
1
Þ!!
when
x
1
\
j
x
j
\
x
2
;
H
ð
x
Þ
¼
:
exp
ð
x
Þ
1
k
x
ð
k
þ
1
=
2
Þ
1
p
1
k¼0
ð
1
Þ
Cð
k
þ
1
=
2
Þ
when
j
x
j
x
2
:
ʦ
The functions
and H are related through the obvious correlation:
ðÞ
¼½1
þ
sin
ðÞ
H
ð
2
1
=
2
U
j
x
jÞ=
2
;
ð
3
:
26
Þ
from which we have:
g
ð
x
Þ
for
x
0
;
Uð
x
Þ
¼
1
g
ð
x
Þ
for
x
0
;
[
where
8
<
:
p
1
2k
1
k
þ
1
j
x
j
1
2
1
=
2
k¼1
ð
1
Þ
for
j
x
j
x
1
;
2
k
1
ð
2k
1
Þð
k
1
Þ!
p
2
Þ
1
k¼0
2k
þ
1
ð
2k
þ
1
Þ!!
j
x
j
1
2
p
exp
ð
x
2
1
=
2
=
for
x
1
\
j
x
j
\
x
2
;
g
ð
x
Þ
¼
p
2
Þ
1
2
p
k
ð
2k
1
Þ!!
j
x
j
1
exp
ð
x
2
=
k¼0
ð
1
Þ
for
j
x
j
x
2
:
p
2k
þ
1
Thus, formula (
3.26
) allows us to calculate W
c
(x) for various values of c and
x. The free parameters x
1
and x
2
in
uence the calculation error. Practically, the
values x
1
= 2.2 and x
2
= 7.5 are acceptable. Moreover, a problem of convergence of
the above rows appears when parameter c is increasing. It is easy to see that
W
c
(x)
fl
1)c
1/2
] for c
≫
.
The formula (
3.13
) is the basic expression for the calculation of W
c
(x). The large
factor, exp(2c), can be normalized by the following expression:
≈ ʦ
[(x
−
1 and W
c
(x)
≈
1
−
exp(
−
cx/2) when x
→
∞
h
i
¼
2
Þ
1
p
c
1
2
k
ð
2k
1
Þ!!
x
2k
exp
ð
2c
x
2
p
exp
ð
2c
ÞU ð
x
þ
1
Þ
=
x
=
k¼0
ð
1
Þ
p
The calculation of W
c
(x) can also be realized by using the Bessel (J) and
Whittaker (
W
) functions:
r
c
2
1
x
l
c
m
J
k
ð
c
ÞW
l
;
s
ð
c
1
2x
Þ
W
c
ð
x
Þ
¼
exp c
ð
1
=
x
Þ;
p
k¼
1
where l =(2k + 3)/4, m =(6k + 9)/4 and s =(1+2k)/4. The following correlations
are very useful as well:
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