
1478 INTEGRAL TRANSFORMS
No.
Laplace transform,
e
f
(p)
Inverse transform, f(x) =
1
2πi
Z
c+i∞
c−i∞
e
px
e
f(p) dp
7
p
p
2
− a
2
−ν−1/2
, ν > 0
a
√
π
(2a)
ν
Γ
ν +
1
2
x
ν
I
ν−1
(ax)
8
(p
2
+ a
2
)
1/2
+ p
−ν
=
a
−2ν
(p
2
+ a
2
)
1/2
− p
ν
, ν > 0
νa
−ν
x
−1
J
ν
(ax)
9
(p
2
− a
2
)
1/2
+ p
−ν
=
a
−2ν
p −(p
2
− a
2
)
1/2
ν
, ν > 0
νa
−ν
x
−1
I
ν
(ax)
10
p
(p
2
+ a
2
)
1/2
+ p
−ν
, ν > 1
νa
1−ν
x
−1
J
ν−1
(ax) −ν(ν + 1)a
−ν
x
−2
J
ν
(ax)
11
p
(p
2
− a
2
)
1/2
+ p
−ν
, ν > 1
νa
1−ν
x
−1
I
ν−1
(ax) − ν(ν + 1)a
−ν
x
−2
I
ν
(ax)
12
p
p
2
+ a
2
+ p
−ν
p
p
2
+ a
2
, ν > −1
a
−ν
J
ν
(ax)
13
p
p
2
− a
2
+ p
−ν
p
p
2
− a
2
, ν > −1
a
−ν
I
ν
(ax)
28.2.5 Expressions with Exponential Fun ctions
No.
Laplace transform,
e
f
(p)
Inverse transform, f(x) =
1
2πi
Z
c+i∞
c−i∞
e
px
e
f(p) dp
1
p
−1
e
−ap
, a > 0
n
0 if 0 < x < a,
1 if a < x
2
p