Tables of Integral Transforms - w3.esfm.ipn.mxw3.esfm.ipn.mx/~cisneros/IntegralTables.pdf · Tables...
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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
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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)
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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)
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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
)
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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)
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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
]
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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)
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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
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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)
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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)
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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
)
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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
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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