Tables of Fourier Sine Transforms T3.4.1.
Trang 1T3.3.4 Expressions with Hyperbolic Functions
∞
0 f(x) cos(ux) dx
2πa–1u
cosh2(ax), a>0 2a2sinhπu1
2πa–1u
3 cosh(ax)
b
cos 1
2πab–1
cosh 12πb–1u
cos πab– 1
+ cosh πb– 1u
cosh(ax) + cos b
πsinh a–1bu
a sin b sinh πa– 1u
5 exp –ax2
2
π
a expb2– u2
4a
cosabu
2
2
4a2cosh2 12πa– 1u
7 sinh(ax)
2b
sin πab–1
cos πab– 1
+ cosh πb– 1u
coth 14πa–1u
T3.3.5 Expressions with Logarithmic Functions
∞
0 f(x) cos(ux) dx
1
ln x if 0< x <1,
1
u Si(u)
2 ln x √
x
– 2π u
ln(4u) + C + π2,
C =0.5772 . is the Euler constant
3 x ν–1ln x, 0< ν <1 Γ(ν) cos 2 u–ν
*
ψ(ν) – π2tan
πν
2
– ln u
+
4 lna + x
a – x
u
cos(au) Si(au) – sin(au) Ci(au)
5 ln 1+ a2/x2
u 1– e–au
6 lna2+ x2
b2+ x2, a, b >0 π u e–bu – e–au
7 e–ax ln x, a>0 – aC+12a ln(u2+ a2) + u arctan(u/a)
u2+ a2
8 ln 1+ e–ax
2usinh πa– 1u
9 ln 1– e–ax
u2 – π
2ucoth πa–1u
Trang 21180 INTEGRALTRANSFORMS
T3.3.6 Expressions with Trigonometric Functions
∞
0 f(x) cos(ux) dx
1 sin(ax)
x , a>0
1
2π if u < a,
1
4π if u = a,
0 if u > a
2 x ν–1sin(ax), a>0, |ν|<1 π (u + a)–ν–|u + a|–ν sign(u – a)
4Γ(1– ν) cos 12πν
3 x sin(ax)
x2+ b2 , a, b >0
1
2πe–ab cosh(bu) if u < a,
–12πe–bu sinh(ab) if u > a
x(x2+ b2), a, b >0
1
2πb–2
1– e–ab cosh(bu)
if u < a,
1
2πb–2e–bu sinh(ab) if u > a
5 e–bx sin(ax), a, b >0 12*(a + u) a + u2+ b2 + a – u
(a – u)2+ b2
+
4ln1–4a2
u2
x2 sin2(ax), a>0 14π(2a – u) if u <2a,
xsin
a
x
xsina √
x
sin b √
x
u sin
ab
2u
sin
a2+ b2
4u –π4
10 sin ax2
a
*
cos
u2
4a
– sin
u2
4a
+
11 exp –ax2
sin bx2
, a>0
√ π (A2+ B2)1/4 exp
2
A2+ B2
sin
ϕ– Bu
2
A2+ B2
,
A=4a, B =4b, ϕ = 12arctan(b/a)
12 1– cos(ax)
2ln1– a2
u2
13 1– cos(ax)
x2 , a>0 12π(a – u) if u < a,
14 x ν–1cos(ax), a>0, 0< ν <1 1
2Γ(ν) cos 1
2πν
|u – a|–ν + (u + a)–ν
15 cos(ax)
x2+ b2 , a, b >0
1
2πb–1e–ab cosh(bu) if u < a,
1
2πb–1e–bu cosh(ab) if u > a
16 e–bx cos(ax), a, b >0 b2*(a + u)12+ b2 + 1
(a – u)2+ b2
+
xcos a √
u sin
a2
4u +π4
xcos a √
x
cos b √
u cos
ab
2u
sin
a2+ b2
4u + π4
19 exp –bx2
b exp
–a
2+ u2
4b
cosh
au
2b
20 cos ax2
a
cos 14a–1u2
+ sin 14a–1u2
21 exp –ax2
cos bx2
, a>0
√ π (A2+ B2)1/4 exp
2
A2+ B2
cos
ϕ– Bu
2
A2+ B2
,
A=4a, B =4b, ϕ = 12arctan(b/a)
Trang 3T3.3.7 Expressions with Special Functions
∞
0 f(x) cos(ux) dx
uarctan
u
a
0 if 0< u < a,
–2π u if a < u
2ulnu + a
u – a
, u≠a
4 J0(ax), a>0
(a2– u2)–1/2 if 0< u < a,
5 J ν (ax), a>0, ν > –1
⎧
⎪
⎪
cos
ν arcsin(u/a)
√
a2– u2 if 0< u < a,
–a
ν sin(πν/2)
ξ(u + ξ) ν if a < u,
where ξ=√
u2– a2
x J ν (ax), a>0, ν >0
⎧
⎨
⎩
ν–1cos
ν arcsin(u/a)
if 0< u < a,
a ν cos(πν/2)
ν u+√
u2– a2 ν if a < u
7 x–ν J ν (ax), a>0, ν > –12
⎧
⎨
⎩
√
π a2– u2 ν–1/2
(2a) νΓν+12 if 0< u < a,
ν+1J
ν (ax),
a>0, –1< ν < –12
⎧
⎨
⎩
2ν+1√
π a ν u
Γ –ν –12
(u2– a2 ν+3/2 if a < u
9 J0 a √
x
usin
a2
4u
x J1 a
√
x
asin2
a2
8u
11 x ν/2J ν a √
x
, a >0, –1< ν < 12 a
2
ν
u–ν–1sin
a2
4u –πν
2
12 J0 a √
x2+ b2
⎧
⎨
⎩
cos b √
a2– u2
√
a2– u2 if 0< u < a,
13 Y0(ax), a>0
0 if 0< u < a, –(u2– a2)–1/2 if a < u
14 x ν Y ν (ax), a>0, |ν|< 12
⎧
⎨
⎩
– (2a) ν √
π
Γ 1
2 – ν
(u2– a2 ν+1/2 if a < u
15 K0 a √
x2+ b2
, a, b >0 2√ π
u2+ a2 exp –b
√
u2+ a2
Trang 41182 INTEGRALTRANSFORMS
T3.4 Tables of Fourier Sine Transforms
T3.4.1 General Formulas
∞
0 f(x) sin(ux) dx
1 af1(x) + bf2(x) a ˇ f1 s(u) + b ˇf2 s(u)
a fsˇu
a
3 x2n (x), n=1,2, (–1)n d
2n
du2n fs(u)ˇ
4 x2n+1f(ax), n=0,1, (–1)n+1 d
2n+1
du2n+1fc(u), ˇˇ fc(u) =
∞
0
f(x) cos(xu) dx
5 f(ax) cos(bx), a, b >0 1
2a
*
ˇ
fs
u + b
a
+ ˇfs
u – b
a
+
T3.4.2 Expressions with Power-Law Functions
∞
0 f(x) sin(ux) dx
1
1 if 0< x < a,
0 if a < x
1
u
1– cos(au)
2
x if 0< x <1,
2– x if 1< x <2,
0 if 2< x
4
u2 sin u sin2 u2
x
π
2
a2+ x2, a>0 π2e–au
x(a2+ x2), a>0 2π
a2 1– e–au
a2+ (x – b)2 –
a
–au sin(bu)
a2+ (x + b)2 –
x – b
–au cos(bu)
9 (x2+ a x 2)n, a>0, n =1,2, πue–au
22n–2(n –1)! a2n–3
n–2
k=0
(2n – k –4)!
k! (n – k –2)!(2au) k
10
x2m+1
(x2+ a) n+1,
n, m =0,1, ;0 ≤m≤n
(–1)n+m2π n!
∂
∂a n a
m e–u √ a
x
π
2u
x √
x
√
2πu
Trang 5No Original function, f (x) Sine transform, ˇfs(u) =
∞
0 f(x) sin(ux) dx
14 a2+ x2– a
1/2
√
a2+ x2
π
2u e
–au
Γ(1– ν)u ν–1
T3.4.3 Expressions with Exponential Functions
∞
0 f(x) sin(ux) dx
a2+ u2
2 x n e–ax, a>0, n =1,2, n! a
a2+ u2
n+1[n/2 ]
k=0
(–1)k C n+2k+11
u
a
2k+1
x e
a
2 (a2+ u2)–3/4sin
3
2arctan
u a
x e
2
a2+ u2– a)1/2
√
a2+ u2
x √
x e
–ax, a>0 √2π a2+ u2– a)1/2
7 x n–1/2e–ax, a>0, n =1,2, (–1)n π2
∂
∂a n
a2+ u2– a 1/2
√
a2+ u2
#
8 x ν–1e–ax, a>0, ν > –1 Γ(ν)(a2+ u2)–ν/2sin
νarctanu
a
9 x–2 e–ax – e–bx
, a, b >0 u
2 ln
u2+ b2
u2+ a2
+ b arctanu
b
– a arctanu
a
e ax+1, a>0
1
2u – π
2a sinh(πu/a)
e ax–1, a>0
π
2a cothπu
a
– 1
2u
12 e x/2
1
2tanh(πu)
4a3/2 uexp
–u
2
4a
2erf
u
2√ a
xexp
–a
x
2u e
–√
2au
cos 2au
+ sin 2au
x √
xexp
–a
x
π
a e
–√
2ausin 2au
Trang 61184 INTEGRALTRANSFORMS
T3.4.4 Expressions with Hyperbolic Functions
∞
0 f(x) sin(ux) dx
a tanh 12πa–1u
4a2cosh2 12πa– 1u
x e
–bx sinh(ax), b>|a| 1
2arctan
2au
u2+ b2– a2
sinh 12πa–1u
5 1– tanh 12ax
asinh πa– 1u
6 coth 12ax
a coth πa–1u
– 1
u
7 cosh(ax)
2b
sinh πb–1u
cos πab– 1
+ cosh πb– 1u
8 sinh(ax)
b
sin 12πab–1
sinh 12πb–1u
cos πab– 1
+ cosh πb– 1u
T3.4.5 Expressions with Logarithmic Functions
∞
0 f(x) sin(ux) dx
1 ln x if 0< x <1,
0 if 1< x
1
u
Ci(u) – ln u – C,
C =0.5772 . is the Euler constant
3 ln x √
2u
ln(4u) + C – π2
–ν
ψ(ν) + π2 cot πν2
– ln u
2Γ(1– ν) cos πν2
5 lna + x
a – x
u sin(au)
6 ln(x + b)
2+ a2 (x – b)2+ a2, a, b >0 2π
u e
–au sin(bu)
7 e–ax ln x, a>0 a arctan(u/a) – 12u ln(u2+ a2) – e C u
u2+ a2
xln 1+ a2x2
–u
a
Trang 7
T3.4.6 Expressions with Trigonometric Functions
∞
0 f(x) sin(ux) dx
1 sin(ax)
2lnu + a
u – a
2 sin(ax)
x2 , a>0
1
2πu if 0< u < a,
1
2πa if u > a
3 x ν–1sin(ax), a>0, –2< ν <1 π |u – a|–ν–|u + a|–ν
4Γ(1– ν) sin 12πν , ν≠ 0
4 sin(ax)
x2+ b2, a, b >0
1
2πb–1e–ab sinh(bu) if 0< u < a,
1
2πb–1e–bu sinh(ab) if u > a
1– x2
sin u if 0< u < π,
0 if u > π
1
a2+ (b – u)2 –
1
a2+ (b + u)2
7 x–1e–ax sin(bx), a>0 14ln(u + b)
2+ a2 (u – b)2+ a2
xsin2(ax), a>0
1
4π if 0< u <2a,
1
8π if u =2a,
0 if u >2a
4(u +2a) ln|u+2a|+14(u –2a) ln|u–2a|–12u ln u
10 exp –ax2
a exp
–u
2+ b2
4a
sinh
bu
2a
x sin(ax) sin(bx), a≥b>0
0
if 0< u < a – b,
π
4 if a – b < u < a + b,
0 if a + b < u
a
x
√ a
2√ u J1 2√ au
xsina
x
8u
sin 2√ au
– cos 2√ au
+ exp –2√ au
x
sin a √
x
8u–3/2exp
–a
2
2u
15 cos(ax)
x , a>0
⎧
⎨
⎩
0 if 0< u < a,
1
4π if u = a,
1
2π if a < u
16 x ν–1cos(ax), a>0, |ν|<1 π(u + a)
–ν – sign(u – a)|u – a|–ν
4Γ(1– ν) cos 12πν
17 x cos(ax)
x2+ b2 , a, b >0
–12πe–ab sinh(bu) if u < a,
1
2πe–bu cosh(ab) if u > a
18 1– cos(ax)
2 lnu2– a2
u2
+ a2lnu + a
u – a
xcos a √
u cos
a2
4u+ π4
xcos a √
x
cos b √
x
, a, b >0 π
ucos
ab
2u
cos
a2+ b2
4u + π4