Embed Size (px)
Citation preview
© 2007 by Taylor & Francis Group, LLC
Appendix B
Tables of Integral Transforms
In this appendix we provide a set of short tables of integral transforms of thefunctions that are either cited in the text or in most common use in math-ematical, physical, and engineering applications. In these tables no attemptis made to give complete lists of transforms. For exhaustive lists of integraltransforms, the reader is referred to Erdelyi 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
f(x) F (k) =1√2π
∞∫
−∞
exp(−ikx)f(x)dx
1 exp(−a|x|), a > 0
(√2π
)a(a2 + k2)−1
2 x exp(−a|x|), a > 0
(√2π
)(−2aik)(a2 + k2)−2
3 exp(−ax2), a > 01√2a
exp(−k2
4a
)
4 (x2 + a2)−1, a > 0√π
2exp(−a|k|)
a
5 x(x2 + a2)−1, a > 0√π
2
(ik
2a
)exp(−a|k|)
6
{c, a≤ x≤ b
0, outside
}ic√2π
1k
(e−ibk − e−iak)
7 |x| exp(−a|x|), a > 0√
2π
(a2 − k2)(a2 + k2)−2
611
© 2007 by Taylor & Francis Group, LLC
612 INTEGRAL TRANSFORMS and THEIR APPLICATIONS
f(x) F (k) =1√2π
∞∫
−∞
exp(−ikx)f(x)dx
8sin ax
x
√π
2H(a− |k|)
9 exp{−x(a− iω)}H(x)1√2π
i
(ω − k + ia)
10 (a2 − x2)− 12 H(a− |x|)
√π
2J0(ak)
11sin[b(x2 + a2) 1
2
]
(x2 + a2) 12
√π
2J0
(a√
b2 − k2)
H(b − |k|)
12cos(b√
a2 − x2)
(a2 − x2) 12
H(a− |x|)√π
2J0
(a√
b2 + k2)
13 e−axH(x), a > 01√2π
(a− ik)(a2 + k2)−1
141√|x|
exp(−a|x|) (a2 + k2)− 12
[a + (a2 + k2) 1
2
] 12
15 δ(x)1√2π
16 δ(n)(x)1√2π
(ik)n
17 δ(x − a)1√2π
exp(−iak)
18 δ(n)(x − a)1√2π
(ik)n exp(−iak)
19 exp(iax)√
2π δ(k − a)
20 1√
2π δ(k)
21 x√
2π i δ′(k)
22 xn√
2π in δ(n)(k)
© 2007 by Taylor & Francis Group, LLC
Tables of Integral Transforms 613
f(x) F (k) =1√2π
∞∫
−∞
exp(−ikx)f(x)dx
23 H(x)√π
2
[1
iπk+ δ(k)
]
24 H(x − a)√π
2
[exp(−ika)
πik+ δ(k)
]
25 H(x) − H(−x)√
2π
(− i
k
)
26 xn exp(iax)√
2π in δ(n)(k − a)
27 |x|−1 1√2π
(A − 2 log |k|), A is a constant
28 log(|x|) −√π
21|k|
29 H(a− |x|)√
2π
(sin ak
k
)
30 |x|α (α< 1, not a negative integer)√
2π
Γ(α+ 1) |k|−(1+α)
× cos[π2
(α+ 1)]
31 sgn x
√2π
1(ik)
32 x−n−1 sgn x1√2π
(−ik)n
n!(A − 2 log |k|)
331x
−i
√π
2sgn k
341xn
−i
√π
2
[(−ik)n−1
(n− 1)!sgn k
]
35 xn exp(iax)√
2π inδ(n)(k − a)
© 2007 by Taylor & Francis Group, LLC
614 INTEGRAL TRANSFORMS and THEIR APPLICATIONS
f(x) F (k) =1√2π
∞∫
−∞
exp(−ikx)f(x)dx
36 xαH(x), (α not an integer)Γ(α+ 1)√
2π|k|−(α+1)
× exp[−(πi
2
)(α+ 1) sgn k
]
37 xn exp(iax)H(x)√
π2
[n!
iπ(k−a)n+1 + in δ(n)(k − a)]
38 exp(iax)H(x − b)√π
2
[exp[−ib(k − a)]
iπ(k − a)+ δ(k − a)
]
391
x− a−i
√π
2exp(−iak)sgn k
401
(x − a)n−i
√π
2exp(−iak)
(−ik)n−1
(n− 1)!sgn k
41eiax
(x − b)i
√π
2exp[ib(a− k)][1 − 2H(k − a)]
42eiax
(x − b)ni
√π
2[1− 2 H(k − a)]
×exp{ib(a− k)}(n− 1)!
[−i(k − a)]n−1
43 |x|α sgn x (α not integer)√
2π
(−i)Γ(α+ 1)|k|α+1
cos(πα
2
)sgn k
44 xn f(x) (−i)n dn
dknF (k)
45dn
dxnf(x) (ik)n F (k)
46 eiax f(bx)1b
F
(k − a
b
)
47 sincos
(ax2) 1√
2a
sincos
(k2
4a − π4
)
© 2007 by Taylor & Francis Group, LLC
Tables of Integral Transforms 615
TABLE B-2 Fourier Cosine Transforms
f(x) Fc(k) =√
2π
∞∫
0
cos(kx)f(x)dx
1 exp(−ax), a > 0
(√2π
)a(a2 + k2)−1
2 x exp(−ax), a > 0
(√2π
)(a2 − k2)(a2 + k2)−2
3 exp(−a2x2)1
|a|√
2exp(− k2
4a2
)
4 H(a− x)√
2π
(sin ak
k
)
5 xa−1, 0 < a < 1√
2π
Γ(a) k−a cos(aπ
2
)
6 cos(ax2)1
2√
a
[cos(
k2
4a
)+ sin
(k2
4a
)]
7 sin(ax2), a > 01
2√
a
[cos(
k2
4a
)− sin
(k2
4a
)]
8 (a2 − x2)v− 12 H(a− x), v >− 1
2 2v− 12 Γ(
v +12
) (a
k
)vJv(ak)
9 (a2 + x2)−1 J0(bx), a, b > 0√
π2 a−1 e−akI0(ab), b < k <∞
10 x−vJv(ax), v >−12
(a2 − k2)v− 12 H(a− k)
2v− 12 av Γ
(v +
12
)
11 (x2 + a2)− 12 e−b(x2+a2)
12 K0
[a(k2 + b2) 1
2
], a > 0, b > 0
12 (2ax− x2)v− 12 H(2a− x), v >− 1
2
√2 Γ(
v +12
)(2a
k
)v
× cos(ak)Jv(ak)
© 2007 by Taylor & Francis Group, LLC
616 INTEGRAL TRANSFORMS and THEIR APPLICATIONS
f(x) Fc(k) =√
2π
∞∫
0
cos(kx)f(x)dx
13 xν−1e−ax, ν > 0, a > 0√
2π Γ(ν)r−ν cos νθ, where
r = (a2 + k2) 12 , θ= tan−1
(ka
)
142x
e−x sin x
√2π
tan−1
(2k2
)
15 sin[a(b2 − x2)
12
]H(b − x)
√π
2(ab)(a2 + k2)−
12
×J1
[b(a2 + k2) 1
2
]
16(1− x2)(1 + x2)2
√π
2k exp(−k)
17 x−α, 0 <α< 1√π
2kα−1
Γ(α)sec(πα
2
)
18(
1a
+ x
)e−ax
√2π
2a2
(a2 + k2)2
19 log(
1 +a2
x2
), a > 0
√2π
(1 − e−ak)k
20 log(
a2 + x2
b2 + x2
), a, b > 0
√2π
(e−bk − e−ak)k
21 a(x2 + a2)−1, a > 0√π
2exp(−ak), k > 0
22 (a2 − x2)−1
√π
2sin(ak)
k
23 e−bx sin(ax)1√2π
[a + k
b2 + (a + k)2+
a− k
b2 + (a− k)2
]
24 e−bx cos(ax)b√2π
[1
b2 + (a− k)2+
1b2 + (a + k)2
]
© 2007 by Taylor & Francis Group, LLC
Tables of Integral Transforms 617
TABLE B-3 Fourier Sine Transforms
f(x) Fs(k) =√
2π
∞∫
0
sin(kx) f(x)dx
1 exp(−ax), a > 0√
2π
k(a2 + k2)−1
2 x exp(−ax), a > 0√
2π
(2ak)(a2 + k2)−2
3 xα−1, 0 <α< 1√
2π
k−αΓ(α) sin(πα
2
)
41√x
1√k
, k > 0
5 xα−1e−ax, α>−1, a > 0√
2π
Γ(α) r−α sin(αθ), where
r = (a2 + k2) 12 , θ= tan−1
(ka
)
6 x−1e−ax, a > 0√
2π
tan−1
(k
a
), k > 0
7 x exp(−a2x2) 2−3/2
(k
a3
)exp(− k2
4a2
)
8 erfc(ax)√
2π
1k
[1 − exp
(− k2
4a2
)]
9 x(a2 + x2)−1
√π
2exp(−ak), a > 0
10 x(a2 + x2)−2 1√2π
(k
a
)exp(−ak), (a > 0)
11 x(a2 − x2)v− 12 H(a− x), 2v− 1
2 av+1k−vΓ(v + 1
2
)
v >− 12 ×Jv+1(ak)
12 tan−1(x
a
) √π
2k−1 exp(−ak)
© 2007 by Taylor & Francis Group, LLC
618 INTEGRAL TRANSFORMS and THEIR APPLICATIONS
f(x) Fs(k) =√
2π
∞∫
0
sin(kx) f(x)dx
13 x−vJv+1(ax), v >−12
k(a2 − k2)v− 12
2v− 12 av+1Γ
(v +
12
) H(a− k)
14 x−1J0(ax)
⎧⎪⎨
⎪⎩
√2π
sin−1
(k
a
), 0 < k < a
√π2 , a < k <∞
⎫⎪⎬
⎪⎭
15 x(a2 + x2)−1 J0(bx), a > 0, b > 0√
π2 e−akI0(ab), a < k <∞
16 J0(a√
x), a > 0√
2π
1k
cos(
a2
4k
)
17 (x2 − a2)v− 12 H(x− a), |v|< 1
2 2v− 12(
ak
)v Γ(v + 1
2
)J−v(ak)
18 x1−v(x2 + a2)−1 Jv(ax),√π
2a−v exp(−ak) Iv(ab),
v >− 32 , a, b > 0 a < k <∞
19 H(a− x), a > 0√
2π
1k
(1 − cos ak)
20 erfc(ax)√
2π
1k
[1 − exp
(− k2
4a2
)]
21 x−α, 0 <α< 2 Γ(1 − α) kα−1 cos(απ
2
)
22 (ax − x2)α− 12 H(a− x), α>− 1
2
√2 Γ(α+
12
)(a
k
)α
× sin(
ak
2
)Jα
(ak
2
)
23 e−bx sin(ax)b√2π
[1
b2 + (a− k)2− 1
b2 + (a + k)2
]
24 ln∣∣∣a+x
b−x
∣∣∣√
2πsin(ak)
k
© 2007 by Taylor & Francis Group, LLC
Tables of Integral Transforms 619
TABLE B-4 Laplace Transforms
f(t) f(s) =∞∫
0
exp(−st) f(t)dt
1 tn (n = 0, 1, 2, 3, . . .)n!
sn+1
2 eat 1s− a
3 cos ats
s2 + a2
4 sinata
s2 + a2
5 coshats
s2 − a2
6 sinhata
s2 − a2
7 tne−at Γ(n + 1)(s + a)n+1
8 ta (a >−1)Γ(a + 1)
sa+1
9 eat cos bts− a
(s − a)2 + b2
10 eat sin btb
(s − a)2 + b2
11 (eat − ebt)a− b
(s − a)(s − b)
121
(a− b)(a eat − bebt)
s
(s − a)(s − b)
13 t sinat2as
(s2 + a2)2
14 t cosats2 − a2
(s2 + a2)2
15 sinat sinh at2sa2
(s4 + 4a4)
© 2007 by Taylor & Francis Group, LLC
620 INTEGRAL TRANSFORMS and THEIR APPLICATIONS
f(t) f(s) =∞∫
0
exp(−st) f(t)dt
16 (sinh at− sin at)2a3
(s4 − a4)
17 (cosh at− cos at)2a2s
(s4 − a4)
18cos at− cos bt
(b2 − a2)(a2 = b2)
s
(s2 + a2)(s2 + b2)
191√t
√π
s
20 2√
t1s
√π
s
21 t coshat (s2 + a2)(s2 − a2)−2
22 t sinh at 2as(s2 − a2)−2
23sin(at)
ttan−1
(a
s
)
24 t−1/2 exp(−a
t
) √π
sexp(−2
√as)
25 t−3/2 exp(−a
t
) √π
aexp(−2
√as)
261√πt
(1 + 2at)eat s
(s− a)√
s− a
27 (1 + at)eat s
(s− a)2
281
2√πt3
(ebt − eat)√
s− a−√
s− b
29 exp(a2t)erf (a√
t)a√
s(s− a2)
30 exp(a2t)erfc (a√
t)1√
s (√
s + a)
311√πt
+ a exp(a2t)erf (a√
t)√
s
(s− a2)
© 2007 by Taylor & Francis Group, LLC
Tables of Integral Transforms 621
f(t) f(s) =∞∫
0
exp(−st) f(t)dt
321√πt
− a exp(a2t) erfc(a√
t)1√
s + a
33exp(−at)√
b − aerf(√
(b − a)t) 1
(s + a)√
s + b
34 12eiωt
[e−λz erfc(ζ −
√iωt) (s− iω)−1 e−z
√sv
+ exp(λz) erfc(ζ +√
iωt)],
where ζ = z/2√
vt, λ=√
iωv .
3512
[e−ab erfc
(b− 2at
2√
t
)e−b(s+a2)
12
+ exp(ab) erfc(
b + 2at
2√
t
)]
36 Si(t) =t∫
0
sinx
xdx
1s
cot−1(s)
37 Ci(t) =−∞∫
t
cosx
xdx − 1
2slog(1 + s2)
38 −Ei(−t)=∞∫
t
e−x
xdx
1s
log(1 + s)
39 J0(at) (s2 + a2)− 12
40 I0(at) (s2 − a2)− 12
41 tα−1 exp(−at), a > 0 Γ(α)(s + a)−α
42√π
Γ(
v +12
)(
t
2a
)v
Jv(at) (s2 + a2)−(v+ 12 ), Re v >− 1
2
43 t−1 Jv(at) av
v(√
s2+a2+s)v , Re v >− 12
44 J0(a√
t)1s
exp(−a2
4s
)
© 2007 by Taylor & Francis Group, LLC
622 INTEGRAL TRANSFORMS and THEIR APPLICATIONS
f(t) f(s) =∞∫
0
exp(−st) f(t)dt
45(
2a
)v
tv/2Jv(a√
t) s−(v+1) exp(−a2
4s
), Re v >− 1
2
46a
2t√πt
exp(−a2
4t
)exp(−a
√s), a > 0
471√πt
exp(−a2
4t
)1√s
exp(−a√
s), a≥ 0
48 exp(−a2t2
4
) √π
aexp(
s2
a2
)erfc
( s
a
), a > 0
49 (t2 − a2)− 12 H(t− a) K0(as), a > 0
50 δ(t− a) exp(−as), a≥ 0
51 H(t− a)1s
exp(−as), a≥ 0
52 δ′(t− a) s e−as, a≥ 0
53 δ(n) (t − a) sn exp(−as)
54 | sin at|, (a > 0)a
(s2 + a2)coth
(πs
2a
)
551√πt
cos(2√
at)1√s
exp(−a
s
)
561√πt
sin(2√
at)1
s√
sexp(−a
s
)
571√πa
cosh(2√
at)1√s
exp(a
s
)
581√πa
sinh(2√
at)1
s√
sexp(a
s
)
59 erf(
t
2a
)1s
exp(a2s2) erfc(as), a > 0
© 2007 by Taylor & Francis Group, LLC
Tables of Integral Transforms 623
f(t) f(s) =∞∫
0
exp(−st) f(t)dt
60 erfc(
a
2√
t
)1s
exp(−a√
s), a≥ 0
61√
4t
πe−
a24t − a erfc
(a
2√
t
)1
s√
sexp(−a
√s), a≥ 0
62 ea(b+at)erfc(
a√
t +b
2√
t
)exp(−b
√s)√
s(√
s + a), a≥ 0
63 J0
(a√
t2 − ω2)
H(t− ω) (s2 + a2)− 12 exp
{−ω
√s2 + a2
}
641t(ebt − eat) log
(s− a
s− b
)
65 {π(t + a)}− 12
1√s
exp(as) erfc(√
as), a > 0
661πt
sin(2a√
t) erf(
a√s
)
671√πt
exp(−2a√
t), a≥ 01√s
exp(
a2
s
)erfc
(a√s
)
68 C(t) =1√2π
t∫
0
cosu√u
du12s
[1√
1 + s2+
s
1 + s2
] 12
69 S(t) =1√2π
t∫
0
sinu√u
du12s
[1√
1 + s2− s
1 + s2
] 12
70 I (t) = 1 + 2∞∑
n=1
exp(−n2πt) (√
s tanh√
s)−1
71 tmα+β−1E(m)α,β (±at) m!sα−β
(sα∓a)m+1
72 1+2at√πt
s+as√
s
