ABSTRACT
J0 ( 2 √ xt ) f (t) dt
21 1 p
˜f ( p +
1 p
) ∫ x 0
J0 ( 2 √ xt – t2
) f (t) dt
22 1
p2ν+1 ˜f ( p +
a
p
) , –
1 2
< ν ≤ 0 ∫ x 0
( x – t at
)ν J2ν ( 2 √ axt – at2
) f (t) dt
23 ˜f (√
p ) ∫ ∞
t
2 √ πx3
exp ( –
t2
4x
) f (t) dt
24 1√ p
˜f (√
p ) 1√
πx
exp ( –
t2
4x
) f (t) dt
25 ˜f ( p +
√ p ) 1
2 √ π
t
(x – t)3/2 exp [ –
t2
4(x – t) ] f (t) dt
26 ˜f (√
p2 + a2 )
f (x) – a ∫ x 0
f (√
x2 – t2 ) J1(at) dt
27 ˜f (√
p2 – a2 )
f (x) + a ∫ x 0
f (√
x2 – t2 ) I1(at) dt
28 ˜f (√
p2 + a2 )√
p2 + a2
J0 ( a √ x2 – t2
) f (t) dt
29 ˜f (√
p2 – a2 )√
p2 – a2
I0 ( a √ x2 – t2
) f (t) dt
30 ˜f (√(p + a)2 – b2 ) e-axf (x) + be-ax ∫ x
0 f (√
x2 – t2 ) I1(bt) dt
31 ˜f (ln p) ∫ ∞ 0
xt-1
Γ(t) f (t) dt
32 1 p
˜f (ln p) ∫ ∞ 0
xt
Γ(t + 1) f (t) dt
33 ˜f (p – ia) + ˜f (p + ia), i2 = –1 2f (x) cos(ax) 34 i
[ ˜f (p – ia) – ˜f (p + ia)], i2 = –1 2f (x) sin(ax)
35 d ˜f (p) dp
–xf (x)
36 d n
˜f (p) dpn
(–x)nf (x)
No Laplace transform, ˜f (p) Inverse transform, f (x) = 1 2πi
epx ˜f (p) dp
37 pn d m
˜f (p) dpm
, m ≥ n (–1)m d n
dxn [ xmf (x)]
38 ∫ ∞ p
˜f (q) dq 1 x f (x)
39 1 p
˜f (q) dq ∫ ∞ x
f (t) t
dt
40 1 p
˜f (q) dq ∫ x 0
f (t) t
dt
No Laplace transform, ˜f (p) Inverse transform, f (x) = 1 2πi
epx ˜f (p) dp
1 1 p
2 1
p + a e-ax
3 1 p2
x
4 1
p(p + a) 1 a
( 1 – e-ax
) 5
1 (p + a)2 xe
6 p
(p + a)2 (1 – ax)e –ax
7 1
p2 – a2 1 a
sinh(ax)
8 p
p2 – a2 cosh(ax)
9 1
(p + a)(p + b) 1
a – b
( e-bx – e-ax
) 10
p
(p + a)(p + b) 1
a – b
( ae-ax – be-bx
) 11
1 p2 + a2
1 a
sin(ax)
12 p
p2 + a2 cos(ax)
13 1
(p + b)2 + a2 1 a e-bx sin(ax)
14 p
(p + b)2 + a2 e –bx [ cos(ax) – b
a sin(ax)
]
No Laplace transform, ˜f (p) Inverse transform, f (x) = 1 2πi
epx ˜f (p) dp
15 1 p3
16 1
p2(p + a) 1 a2 ( e-ax + ax – 1
) 17
1 p(p + a)(p + b)
1 ab(a – b)
( a – b + be-ax – ae-bx
) 18
1 p(p + a)2
1 a2 ( 1 – e-ax – axe-ax
) 19
1 (p + a)(p + b)(p + c)
(c – b)e-ax + (a – c)e-bx + (b – a)e-cx (a – b)(b – c)(c – a)
20 p
(p + a)(p + b)(p + c) a(b – c)e-ax + b(c – a)e-bx + c(a – b)e-cx
(a – b)(b – c)(c – a)
21 p2
(p + a)(p + b)(p + c) a2(c – b)e-ax + b2(a – c)e-bx + c2(b – a)e-cx
(a – b)(b – c)(c – a) 22
1 (p + a)(p + b)2
1 (a – b)2
[ e-ax – e-bx + (a – b)xe-bx]
23 p
(p + a)(p + b)2 1
(a – b)2 { –ae-ax + [a + b(b – a)x]e-bx}
24 p2
(p + a)(p + b)2 1
(a – b)2 [ a2e-ax + b(b – 2a – b2x + abx)e-bx]
25 1
(p + a)3 1 2x
26 p
(p + a)3 x ( 1 – 12ax
) e-ax
27 p2
(p + a)3 ( 1 – 2ax + 12 a
2x2 ) e-ax
28 1
p(p2 + a2) 1 a2 [ 1 – cos(ax)]
29 1
p [(p + b)2 + a2] 1a2 + b2
{ 1 – e-bx
[ cos(ax) + b
a sin(ax)
]} 30
1 (p + a)(p2 + b2)
1 a2 + b2
[ e-ax +
a
b sin(bx) – cos(bx)
] 31
p
(p + a)(p2 + b2) 1
a2 + b2 [ –ae-ax + a cos(bx) + b sin(bx)]
32 p2
(p + a)(p2 + b2) 1
a2 + b2 [ a2e-ax – ab sin(bx) + b2 cos(bx)]
33 1
p3 + a3
1 3a2 e
–ax –
1 3a2 e
ax/2[cos(kx) –√3 sin(kx)], k = 12 a
√ 3
34 p
p3 + a3 –
1 3a e
–ax + 1 3a e
ax/2[cos(kx) +√3 sin(kx)], k = 12 a
√ 3
No Laplace transform, ˜f (p) Inverse transform, f (x) = 1 2πi
epx ˜f (p) dp
35 p2
p3 + a3 1 3 e
–ax + 23 e ax/2 cos(kx), k = 12a
√ 3
36 1(
p + a)[(p + b)2 + c2] e-ax – e-bx cos(cx) + ke-bx sin(cx)(a – b)2 + c2 , k = a – bc 37
p( p + a)[(p + b)2 + c2]
–ae-ax + ae-bx cos(cx) + ke-bx sin(cx) (a – b)2 + c2 ,
k = b2 + c2 – ab
c
38 p2(
p + a)[(p + b)2 + c2] a2e-ax+(b2 +c2 –2ab)e-bx cos(cx)+ke-bx sin(cx)
(a-b)2 +c2 ,
k = –ac – bc + ab2 – b3
c
39 1 p4
40 1
p3(p + a) 1 a3
–
1 a2
x + 1 2a
x2 – 1 a3
e-ax
41 1
p2(p + a)2 1 a2
x ( 1 + e-ax
) +
2 a3 ( e-ax – 1
) 42
1 p2(p + a)(p + b) –
a + b
a2b2 +
1 ab
x + 1
a2(b – a) e –ax +
1 b2(a – b) e
43 1
(p + a)2(p + b)2 1
(a – b)2 [ e-ax
( x +
2 a – b
) + e-bx
( x –
2 a – b
)] 44
1 (p + a)4
45 p
(p + a)4 1 2x
46 1
p2(p2 + a2) 1 a3 [ ax – sin(ax)]
47 1
p4 – a4 1
2a3 [ sinh(ax) – sin(ax)]
48 p
p4 – a4 1
2a2 [ cosh(ax) – cos(ax)]
49 p 2
p4 – a4 1 2a [ sinh(ax) + sin(ax)]
50 p3
p4 – a4 1 2 [ cosh(ax) + cos(ax)]
51 1
p4 + a4 1
a3 √
2 ( cosh ξ sin ξ – sinh ξ cos ξ
) , ξ =
ax√ 2
52 p
p4 + a4 1 a2
sin ( ax√
) sinh
( ax√ 2
) 53
p2
p4 + a4 1
a √
2 ( cos ξ sinh ξ + sin ξ cosh ξ
) , ξ =
ax√ 2
No Laplace transform, ˜f (p) Inverse transform, f (x) = 1 2πi
epx ˜f (p) dp
54 1
(p2 + a2)2 1
2a3 [ sin(ax) – ax cos(ax)]
55 p
(p2 + a2)2 1 2a
x sin(ax)
56 p2
(p2 + a2)2 1 2a [ sin(ax) + ax cos(ax)]
57 p3
(p2 + a2)2 cos(ax) – 1 2 ax sin(ax)
58 1[(p + b)2 + a2]2 12a3 e-bx[sin(ax) – ax cos(ax)]
59 1
(p2 – a2)(p2 – b2) 1
a2 – b2
[ 1 a
sinh(ax) – 1 b sinh(bx)
] 60
p
(p2 – a2)(p2 – b2) cosh(ax) – cosh(bx)
a2 – b2
61 p2
(p2 – a2)(p2 – b2) a sinh(ax) – b sinh(bx)
a2 – b2
62 p3
(p2 – a2)(p2 – b2) a2 cosh(ax) – b2 cosh(bx)
a2 – b2
63 1
(p2 + a2)(p2 + b2) 1
b2 – a2
[ 1 a
sin(ax) – 1 b sin(bx)
] 64
p
(p2 + a2)(p2 + b2) cos(ax) – cos(bx)
b2 – a2
65 p2
(p2 + a2)(p2 + b2) –a sin(ax) + b sin(bx)
b2 – a2
66 p3
(p2 + a2)(p2 + b2) –a2 cos(ax) + b2 cos(bx)
b2 – a2
67 1 pn
, n = 1, 2, . . . 1(n – 1)!x n-1
68 1
(p + a)n , n = 1, 2, . . . 1
(n – 1)!x n-1e-ax
69 1
p(p + a)n , n = 1, 2, . . . a –n [ 1 – e-axen(ax)
] , en(z) = 1 + z1! + · · · +
zn
n!
