Adjustment
209
For the inverse Laplace transformation, we use
L
−1 (s
−n e
−a/s )=(
t
a
)
(n−1)/2 J n−1 (2
√
at),
(9.51)
where J n−1 is the Bessel function of the first kind of order n − 1.
◮
Example 9.1: Bessel functions
The Bessel functions J n (x) are defined by
J n (x)=
∞
k=0
(−1) k
k!(n + k)!
(
x
2
)
n+2k ,
(9.52)
and satisfy the differential equation
x
2 y
′′ + xy
′ +(x
2 − n
2 )y =0.
(9.53)
The functions J 0 ,J 1 and J 2 are plotted in Fig. 9.5. As can be seen, these funcFigure 9.5. Bessel functions J0(x), J1(x) and J2(x) as a function of x.
tions oscillate for x →∞ . The asymptotic behavior for x → 0 and x →∞
is
J n (x) →
2
πx
cos(x −
π
4
−
nπ
2
)+O(
1
x
),x →∞,
J n (x) →
1
n!
(
x
2
)
n ,x → 0,
◭
209
For the inverse Laplace transformation, we use
L
−1 (s
−n e
−a/s )=(
t
a
)
(n−1)/2 J n−1 (2
√
at),
(9.51)
where J n−1 is the Bessel function of the first kind of order n − 1.
◮
Example 9.1: Bessel functions
The Bessel functions J n (x) are defined by
J n (x)=
∞
k=0
(−1) k
k!(n + k)!
(
x
2
)
n+2k ,
(9.52)
and satisfy the differential equation
x
2 y
′′ + xy
′ +(x
2 − n
2 )y =0.
(9.53)
The functions J 0 ,J 1 and J 2 are plotted in Fig. 9.5. As can be seen, these funcFigure 9.5. Bessel functions J0(x), J1(x) and J2(x) as a function of x.
tions oscillate for x →∞ . The asymptotic behavior for x → 0 and x →∞
is
J n (x) →
2
πx
cos(x −
π
4
−
nπ
2
)+O(
1
x
),x →∞,
J n (x) →
1
n!
(
x
2
)
n ,x → 0,
◭