Eduardo Kausel-Fundamental solutions in elastodynamics_ a compendium-Cambridge University Press (2006)

196 Brief listing of integral transforms
∫
1 ∞
2π
−∞
⎧
e ±i k x ⎪⎨
k ( )
k 2 − k0
2 dk =
⎪⎩
∫
1 ∞
e ±i k x
(
2π k2 − k0) 2 2
dk =
−∞
∫
1 ∞
ke ±i k x
(
2π k2 − k0) 2 2
dk =
−∞
⎧
⎪⎨
⎪⎩
⎧
⎪⎨
± i (
e
−i k0|x| − 2 ) sgn(x) Imk 0 < 0
2 k0
2
± i ( cos k 0 x − 2) sgn(x) Im k 0 = 0
2 k0
2
− 1 (k 0 |x| − i ) e −i k0|x| Im k 0 < 0
4k 3 0
− 1 |x| cos k 0 x Im k 0 = 0
4k0
2
± x
4k 0
e −i k0|x| Im k 0 < 0
⎪⎩ ± i (cos k 0 x − k 0 x sin k 0 x) sgn x Im k 0 = 0
4k 2 0
⎧
1
∫
1 ∞
k 2 e ±i k x ⎪⎨ (1 − i k 0 |x|) e −i k0|x| Im k 0 < 0
(
2π −∞ k2 − k0) 2 2
dk =
4i k 0
|x| ⎪⎩
4 cos k 0x Im k 0 = 0
∫
1 ∞
2π
−∞
⎧⎪ 1
e ±i k x
⎨
(
k2 − k ) 1/ dk = 0
2 2 ⎪ ⎩
2i H(2)
0
(k 0 |x|) Imk 0 < 0
1
i J 0 (k 0 |x|) Im k 0 = 0
⎧
∫
1 ∞
ke ±i k x ⎨
( )
2π −∞
1/ dk = ± 1 2 k 0 H (2)
1
(k 0 |x|) sgn(x) Im k 0 < 0
k2 − k0
2 2 ⎩
±k 0 J 1 (k 0 |x|) sgn(x) Im k 0 = 0
⎧⎪ ∫ i |x|
1 ∞
e ±i k x ⎨ H (2)
(
2π −∞ k2 − k ) 3/ dk = 1
(k 0 |x|) Im k 0 < 0
2 k 0
0
2 2 ⎪ i |x| ⎩ J 1 (k 0 |x|) Im k 0 = 0
k 0
⎧
∫
1 ∞
ke ±i k x ⎨
( )
2π −∞
3/ dk = ± 1 2 x H(2) 0
(k 0 |x|) Im k 0 < 0
k2 − k0
2 2 ⎩
±x J 0 (k 0 |x|) Im k 0 = 0
12.2 Hankel transforms
The Hankel transform is defined by the integral
F n (k) =
∫ ∞
0
r ν f (r) J n (kr) dr
in which ν is defined in various references as either 1/2 or1. Since the Bessel function
behaves asymptotically as a harmonic function while its amplitude decays as r −1/2 , the
Hankel transform exists only if the integral ∫ ∞
∣
0 r ν−1/2 f (r) ∣ dr exists (i.e., if the Dirichlet