Appendix B
Tables of Integral Transforms
In this appendix we provide a set of short tables of integral transforms of the
functions that are either cited in the text or in most common use in math-
ematical, physical, and engineering applications. In these tables no attempt
is made to give complete lists of transforms. For exhaustive lists of integral
transforms, the reader is referred to Erdélyi et al. (1954), Campbell and Foster
(1948), Ditkin and Prudnikov (1965), Doetsch (1950 1956, 1970), Marichev
(1983), and Oberhettinger (1972, 1974).
TABLE B-1 Fourier Transforms
"
1
f(x) F (k) = " exp(-ikx)f(x)dx
2Ä„
-"

2
1 exp(-a|x|), a > 0 a(a2 + k2)-1
Ä„

2
2 x exp(-a|x|), a > 0 (-2aik)(a2 + k2)-2
Ä„

1 k2
3 exp(-ax2), a > 0 " -
exp
4a
2a

Ä„ exp(-a|k|)
4 (x2 + a2)-1, a > 0
2 a

Ä„ ik
5 x(x2 + a2)-1, a > 0 exp(-a|k|)
2 2a

c, ad"xd"b
ic 1
6 " (e-ibk-e-iak)
k
0, outside 2Ä„

2
7 |x|exp(-a|x|), a > 0 (a2-k2)(a2 + k2)-2
Ä„
611
© 2007 by Taylor & Francis Group, LLC
612 INTEGRAL TRANSFORMS and THEIR APPLICATIONS
"
1
f(x) F (k) = " exp(-ikx)f(x)dx
2Ä„
-"

sin ax Ä„
8 H(a-|k|)
x 2
1 i
9 exp{-x(a-iÉ)}H(x) "
(É-k + ia)
2Ä„

1 Ä„
10 (a2-x2)- 2
H(a-|x|) J0(ak)
2

1

2

sin b(x2 + a2)
Ä„
11 J0 a b2-k2 H(b-|k|)
1
2 2
(x2 + a2)
"



cos b a2-x2
Ä„
12 H(a-|x|) J0 a b2 + k2
1
2 2
(a2-x2)
1
13 e-axH(x), a > 0 " (a-ik)(a2 + k2)-1
2Ä„
1

1 1 1 2
14 exp(-a|x|) (a2 + k2)- 2 2
a + (a2 + k2)
|x|
1
15 ´(x) "
2Ä„
1
16 ´(n)(x) " (ik)n
2Ä„
1
17 ´(x-a) " exp(-iak)
2Ä„
1
18 ´(n)(x-a) " (ik)n exp(-iak)
2Ä„
"
19 exp(iax) 2Ä„ ´(k-a)
"
20 1 2Ä„ ´(k)
"
21 x 2Ä„ i ´2 (k)
"
22 xn 2Ä„ in ´(n)(k)
© 2007 by Taylor & Francis Group, LLC
Tables of Integral Transforms 613
"
1
f(x) F (k) =" exp(-ikx)f(x)dx
2Ä„
-"


Ä„ 1
23 H(x) + ´(k)
2 iĄk

Ä„ exp(-ika)
24 H(x-a) + ´(k)
2 Ä„ik


2 i
25 H(x)-H(-x) -
Ä„ k
"
26 xn exp(iax) 2Ä„ in ´(n)(k-a)
1
27 |x|-1 " (A-2 log|k|), A is a constant
2Ä„

Ä„ 1
28 log(|x|) -
2|k|


2 sin ak
29 H(a-|x|)
Ä„ k

2
30 |x|Ä… (Ä… < 1, not a negative integer) “(Ä… + 1)|k|-(1+Ä…)
Ä„
Ä„
×cos (Ä… + 1)
2

2 1
31 sgn x
Ä„ (ik)
1 (-ik)n
32 x-n-1 sgn x " (A-2 log|k|)
n!
2Ä„

1 Ä„
33 -i sgn k
x 2

1 Ä„ (-ik)n-1
34 -i sgn k
xn 2 (n-1)!
"
35 xn exp(iax) 2Ä„ in´(n)(k-a)
© 2007 by Taylor & Francis Group, LLC
614 INTEGRAL TRANSFORMS and THEIR APPLICATIONS
"
1
f(x) F (k) =" exp(-ikx)f(x)dx
2Ä„
-"
“(Ä… + 1)
36 xÄ…H(x), (Ä… not an integer) " |k|-(Ä…+1)
2Ä„

Ä„i
×exp - (Ä… + 1) sgn k
2


Ä„ n!
37 xn exp(iax) H(x) + in ´(n)(k-a)
2 iĄ(k-a)n+1


Ä„ exp[-ib(k-a)]
38 exp(iax) H(x-b) + ´(k-a)
2 iĄ(k-a)

1 Ä„
39 -i exp(-iak)sgn k
x-a 2

1 Ä„ (-ik)n-1
40 -i exp(-iak) sgn k
(x-a)n 2 (n-1)!

eiax Ä„
41 i exp[ib(a-k)][1-2H(k-a)]
(x-b) 2

eiax Ä„
42 i [1-2 H(k-a)]
(x-b)n 2
exp{ib(a-k)}
× [-i(k-a)]n-1
(n-1)!

Ä„Ä…
2 (-i)“(Ä… + 1)
43 |x|Ä… sgn x (Ä… not integer) cos sgn k
Ä„ |k|Ä…+1 2
dn
44 xn f(x) (-i)n F (k)
dkn
dn
45 f(x) (ik)n F (k)
dxn

1 k-a
46 eiax f(bx) F
b b


1
sin sin
k2
47 ax2 " -Ä„
4a 4
cos cos
2a
© 2007 by Taylor & Francis Group, LLC