ABSTRACT
Jp(at) ˜f (t) dt
14 f (x+ a) = f (x) (periodic function) 11 – eap ∫ a 0
f (x)e-px dx
15 f (x+ a) = –f (x)(antiperiodic function) 1
1 + e-ap
f (x)e-px dx
16 f ′x(x) p ˜f (p) – f (+0)
17 f (n)x (x) pn ˜f (p) – n∑
pn-kf (k-1)x (+0)
No Original function, f (x) Laplace transform, ˜f (p) = ∫ ∞ 0
e-pxf (x) dx
18 xmf (n)x (x), m ≥ n ( –
d
dp
)m[ pn ˜f (p)]
19 d n
dxn [ xmf (x)], m ≥ n (–1)mpn dm
dpm ˜f (p)
20 ∫ x 0
f (t) dt ˜f (p) p
21 ∫ x 0
(x – t)f (t) dt 1 p2
˜f (p)
22 ∫ x 0
(x – t)νf (t) dt, ν > –1 Γ(ν + 1)p-ν-1 ˜f (p)
23 ∫ x 0
e-a(x-t)f (t) dt 1 p + a
˜f (p)
24 ∫ x 0
sinh [ a(x – t)]f (t) dt a ˜f (p)
p2 – a2
25 ∫ x 0
sin [ a(x – t)]f (t) dt a ˜f (p)
p2 + a2
26 ∫ x 0
f1(t)f2(x – t) dt ˜f1(p) ˜f2(p)
27 ∫ x 0
1 t f (t) dt 1
p
˜f (q) dq
28 ∫ ∞ x
1 t f (t) dt 1
p
˜f (q) dq
29 ∫ ∞ 0
1√ t sin ( 2 √ xt ) f (t) dt
√ π
p √ p
˜f ( 1 p
)
30 1√ x
cos ( 2 √ xt ) f (t) dt
√ π√ p
˜f ( 1 p
) 31
1√ πx
exp ( –
t2
4x
) f (t) dt 1√
p ˜f (√
p )
32 ∫ ∞ 0
t
2 √ πx3
exp ( –
t2
4x
) f (t) dt ˜f(√p)
33 f (x) – a ∫ x 0
f (√
x2 – t2 ) J1(at) dt ˜f
(√ p2 + a2
) 34 f (x) + a
f (√
x2 – t2 ) I1(at) dt ˜f
(√ p2 – a2
)
No Original function, f (x) Laplace transform, ˜f (p) = ∫ ∞ 0
e-pxf (x) dx
1 1 1 p
{ 0 if 0 < x < a, 1 if a < x < b, 0 if b < x.
