LAPLACE TRANSFORMS 181

Table of General Properties of Laplace Transforms

f (s) F(t)

33.3. a f1 (s) + bf2 (s) aF1 (t ) + bF2 (t )

33.4. f (s /a) a F (at )

33.5. f (s – a) eatF(t)

33.6. e–asf (s) (t − a) = { F (t − a) t > a

0 t<a

33.7. sf (s) – F (0) F ′ (t )

33.8. s 2 f (s) − sF (0) − F ′(0) F ′′(t )

33.9. s n f (s) − s n−1 F (0) − s n− 2 F ′(0) −  − F ( n−1) (0) F(n)(t)

33.10. f ′ (s ) –tF(t)

33.11. f ′′(s) t2F(t)

33.12. f (n)(s) (–1)nt nF(t)

f (s ) t
33.13.
s ∫ 0
F (u)du

f (s ) t t (t − u)n−1
t
33.14.
sn ∫ 0
 ∫ F (u)du n =
0 ∫0 (n − 1)! F (u)du

t
33.15. f (s)g(s) ∫ 0
F (u)G (t − u)du
182 LAPLACE TRANSFORMS

f (s) F(t)

∞ F (t )
33.16. ∫ s
f (u)du
t

1 T
33.17.
1 − e − sT ∫ 0
e − su F (u)du F(t) = F(t + T)

f ( s) 1 ∞
∫ e−u
2
/ 4t
33.18. F (u)du
s πt 0

33.19.
1 1
f
s s () ∫
∞

0
J 0 (2 ut )F (u)du

33.20.
s
1
n+1 f (1s) 0
∞
t n / 2 ∫ u − n / 2 J n (2 ut )F (u)du

f (s + 1/s) t
33.21.
s2 + 1 ∫ 0
J 0 (2 u(t − u)) F (u)du

1 ∞
∫ u −3 / 2 e − s
2
/ 4u
33.22. f (u)du F(t2)
2 π 0

f (ln s) ∞ t u F (u)
33.23.
s ln s ∫ 0 Γ (u + 1)
du

P (s ) n
P(α k ) α t
33.24. Q (s ) ∑ Q′(α
k =1 k)
e k

P(s) = polynomial of degree less than n,

Q(s) = (s – a1)(s – a2) … (s – an)
where a1, a2, …, an are all distinct.
LAPLACE TRANSFORMS 183

Table of Special Laplace Transforms

f (s) F(t)

1
33.25. 1
s

1
33.26. t
s2

1 t n−1
33.27. n = 1, 2, 3,… , 0! = 1
sn (n − 1)!

1 t n−1
33.28. n>0
sn Γ(n)

1
33.29. eat
s−a

1 t n−1e at
33.30. n = 1, 2, 3,… , 0! = 1
(s − a) n (n − 1)!

1 t n−1e at
33.31. n>0
(s − a) n Γ(n)

1 sin at
33.32.
s + a2
2
a

s
33.33. cos at
s2 + a2

1 e bt sin at
33.34.
(s − b) 2 + a 2 a

s−b
33.35. e bt cos at
(s − b) 2 + a 2

1 sinh at
33.36.
s2 − a2 a

s
33.37. cosh at
s2 − a2

1 e bt sinh at
33.38.
(s − b) 2 − a 2 a
184 LAPLACE TRANSFORMS

f (s) F(t)
s−b
33.39. e bt cosh at
(s − b) 2 − a 2
1
a≠b e bt − e at
33.40. (s − a)(s − b) b−a
s
a≠b be bt − ae at
33.41. (s − a)(s − b) b−a
1 sin at − at cos at
33.42. (s 2 + a 2 ) 2 2a 3
s t sin at
33.43. (s 2 + a 2 ) 2 2a
s2 sin at + at cos at
33.44.
(s + a 2 ) 2
2
2a
s3
33.45. cos at − 12 at sin at
(s + a 2 ) 2
2

s2 − a2
33.46. t cos at
(s 2 + a 2 ) 2
1 at cosh at − sinh at
33.47. (s 2 − a 2 ) 2 2a 3
s t sinh at
33.48. (s 2 − a 2 ) 2 2a
s2 sinh at + at cosh at
33.49.
(s 2 − a 2 ) 2 2a
s3
33.50. cosh at + 12 at sinh at
(s − a 2 ) 2
2

s2
33.51. t cosh at
(s − a 2 ) 3 / 2
2

1 (3 − a 2 t 2 ) sin at − 3at cos at

33.52. (s + a 2 ) 3
2
8a 5
s t sin at − at 2 cos at
33.53. (s + a 2 )3
2
8a 3
s2 (1 + a 2 t 2 ) sin at − at cos at
33.54.
(s + a 2 )3
2
8a 3
s3 3t sin at + at 2 cos at
33.55.
(s 2 + a 2 )3 8a
LAPLACE TRANSFORMS 185

f (s) F(t)

s4 (3 − a 2 t 2 ) sin at + 5at cos at

33.56.
(s + a 2 )3
2
8a
s5 (8 − a 2 t 2 ) cos at − 7at sin at
33.57.
(s + a 2 )3
2
8
3s 2 − a 2 t 2 sin at
33.58.
(s 2 + a 2 )3 2a
s 3 − 3a 2 s
33.59. 1
t 2 cos at
(s 2 + a 2 )3 2

s 4 − 6a 2 s 2 + a 4
33.60. 1
t 3 cos at
(s 2 + a 2 ) 4 6

s3 − a2s t 3 sin at
33.61.
(s 2 + a 2 ) 4 24 a
1 (3 + a 2 t 2 ) sinh at − 3at cosh at
33.62. (s − a 2 ) 3
2
8a 5
s at 2 cosh at − t sinh at
33.63. (s − a 2 ) 3
2
8a 3
s2 at cosh at + (a 2 t 2 − 1) sinh at
33.64.
(s − a 2 ) 3
2
8a 3
s3 3t sinh at + at 2 cosh at
33.65.
(s − a 2 )3
2
8a
s4 (3 + a 2 t 2 ) sinh at + 5at cosh at
33.66.
(s − a 2 )3
2
8a
s5 (8 + a 2 t 2 ) cosh at + 7at sinh at
33.67.
(s − a 2 )3
2
8
3s 2 + a 2 t 2 sinh at
33.68.
(s 2 − a 2 )3 2a
s 3 + 3a 2 s
33.69. 1
t 2 cosh at
(s 2 − a 2 )3 2

s 4 + 6a 2 s 2 + a 4
33.70. 1
t 3 cosh at
(s 2 − a 2 ) 4 6

s3 + a2s t 3 sinh at
33.71.
(s 2 − a 2 ) 4 24 a

1 e at / 2 ⎧ 3at 3at ⎫
33.72. 2 ⎨ 3 sin − cos + e −3at / 2 ⎬
s + a3
3 3a ⎩ 2 2 ⎭
186 LAPLACE TRANSFORMS

f (s) F(t)

s e at / 2 ⎧ 3at 3at ⎫
+ 3 sin − e −3at / 2 ⎬
3a ⎨⎩
33.73. cos
s + a3
3 2 2 ⎭

s2 1 ⎛ − at 3at ⎞
e + 2e at / 2 cos
33.74.
s3 + a3 3 ⎜⎝ 2 ⎟⎠

1 e − at / 2 ⎧ 3at / 2 3at 3at ⎫

− cos − 3 sin
33.75.
s − a3
3 3a 2 ⎨e
⎩ 2 2 ⎬⎭

s e − at / 2 ⎧ 3at 3at ⎫
− cos + e3at / 2 ⎬
3a ⎨⎩
33.76. 3 sin
s − a3
3 2 2 ⎭

s2 1 ⎛ at 3at ⎞
e + 2e − at / 2 cos
33.77.
s − a3
3 3 ⎜⎝ 2 ⎟⎠

1 1
33.78. (sin at cosh at − cos at sinh at )
s 4 + 4a 4 4a3
s sin at sinh at
33.79.
s 4 + 4a 4 2a 2
s2 1
33.80. (sin at cosh at + cos at sinh at )
s + 4a 4
4
2a
s3
33.81. cos at cosh at
s + 4a 4
4

1 1
33.82. (sinh at − sin at )
s4 − a4 2a 3
s 1
33.83. (cosh at − cos at )
s − a4
4
2a 2
s2 1
33.84. (sinh at + sin at )
s − a4
4
2a
s3 1
(cosh at + cos at )
33.85.
s − a4
4 2

1 e − bt − e − at
33.86.
s+a + s+b 2(b − a) π t 3

1 erf at
33.87.
s s+a a
1 e at erf at
33.88.
s (s − a) a
1 ⎧ 1 ⎫
e at ⎨ − b e b t erfc(b t )⎬
2
33.89.
s−a +b ⎩ πt ⎭
LAPLACE TRANSFORMS 187

f (s) F(t)

1
33.90. J 0 (at )
s2 + a2

1
33.91. I 0 (at )
s2 − a2

( s 2 + a 2 − s) n
33.92. n > −1 a n J n (at )
s2 + a2

(s − s 2 − a 2 ) n
33.93. n > −1 a n I n (at )
s2 − a2

e b(s− s +a )
2 2

33.94. J 0 (a t (t + 2b))
s2 + a2

e− b s +a ⎧J 0 (a t 2 − b 2 ) t > b
2 2

33.95. ⎨
s2 + a2 ⎩0 t<b

1 tJ1 (at )
33.96. (s 2 + a 2 ) 3 / 2 a

s
33.97. tJ 0 (at )
(s 2 + a 2 )3 / 2

s2
33.98. J 0 (at ) − atJ1 (at )
(s + a 2 ) 3 / 2
2

1 tI1 (at )
33.99. (s − a 2 )3 / 2
2
a

s
33.100. tI 0 (at )
(s 2 − a 2 ) 3 / 2

s2
33.101. I 0 (at ) + atI1 (at )
(s − a 2 ) 3 / 2
2

1 e− s
= F (t ) = n, n  t < n + 1, n = 0,1, 2,…
33.102. s(e − 1) s(1 − e − s )
s

See also entry 33.165.

[t ]
1 e− s F (t ) = ∑ r k
33.103. =
s(e − r ) s(1 − re − s )
s k =1

where [t] = greatest integer  t

es − 1 1 − e− s
=
33.104. s(e − r ) s(1 − re − s )
s F (t ) = r n , n  t < n + 1, n = 0,1, 2,…
See also entry 33.167.
e − a /s cos 2 at
33.105.
s πt
188 LAPLACE TRANSFORMS

f (s) F(t)

e − a /s sin 2 at
33.106.
s3/ 2 πa
n/2
e − a /s ⎛ t⎞
33.107. n > −1 J n (2 at )
s n+1 ⎝ a⎠

e− a s e− a / 4t
2

33.108.
s πt

a
e− a
2
/ 4t
33.109. e− a s
2 π t3

1 − e− a s
33.110. erf (a / 2 t )
s

e− a s
33.111. erfc(a / 2 t )
s

e− a s ⎛ a ⎞
e b ( bt + a ) erfc ⎜ b t +
2 t ⎟⎠
33.112.
s ( s + b) ⎝

e− a / s 1 ∞
∫ une−u
2
/ 4 a2t
33.113. n > −1 J 2 n (2 u )du
s n+1 π ta 2 n+1 0

⎛ s + a⎞ e − bt − e − at
33.114. ln
⎝ s + b⎠ t

ln[(s 2 + a 2 ) /a 2 ] Ci(at )
33.115.
2s
ln[(s + a) /a]
33.116. Ei(at )
s
(γ + ln s)
−
33.117. s ln t
γ = Euler’s constant = .5772156 …
⎛ s2 + a2 ⎞ 2(cos at − cos bt )
ln ⎜ 2
⎝ s + b 2 ⎟⎠
33.118.
t

π 2 (γ + ln s)2
+ ln 2 t
33.119. 6s s
γ = Euler’s constant = .5772156 …
ln s −(ln t + γ )
33.120.
s γ = Euler’s constant = .5772156 …
ln 2 s (ln t + γ )2 − 16 π 2
33.121.
s γ = Euler’s constant = .5772156 …
LAPLACE TRANSFORMS 189

f (s) F(t)

Γ ′(n + 1) − Γ (n + 1) ln s
33.122. n > −1 t n ln t
s n+1

sin at
33.123. tan −1 (a /s)
t

tan −1 (a /s) Si(at )

33.124.
s

e a /s e −2 at
33.125. erfc( a /s)
s πt

2
/ 4 a2
2a − a t 2 2

33.126. es erfc(s / 2a) e

π
2
/ 4 a2
es erfc(s / 2a) erf(at )
33.127.
s

e as erfc as 1
33.128.
s π (t + a)

1
33.129. e as Ei(as)
t+a

33.130.
1⎡
a ⎢⎣
cos as
π
2 { ⎤
− Si(as) − sin as Ci(as)⎥
⎦
} 1
t 2 + a2

33.131. sin as {
π
2
− Si(as) + cos as Ci(as) } t
t 2 + a2

33.132.
cos as { π
2 }
− Si(as) − sin as Ci(as) tan −1 (t /a)
s

33.133.
sin as {
π
2
− Si(as) − cos as Ci(as) } 1 ⎛ t 2 + a2 ⎞
2 ⎜⎝ a 2 ⎟⎠
ln
s

⎡π − Si(as)⎤ + Ci 2 (as)
2
1 ⎛ t 2 + a2 ⎞
t ⎜⎝ a 2 ⎟⎠
33.134. ln
⎢⎣ 2 ⎥⎦

33.135. 0 (t) = null function

33.136. 1 δ (t) = delta function

33.137. e − as δ (t − a)

e − as
33.138. s (t − a)
See also entry 33.163.
190 LAPLACE TRANSFORMS

f (s) F(t)

x 2 ∞ (−1)n nπ x nπ t
a π∑
sinh sx + sin cos
33.139. n a a
s sinh sa n =1

4 ∞ (−1)n (2n − 1)π x (2n − 1)π t

π∑
sinh sx sin sin
33.140. 2 n − 1 2 a 2a
s cosh sa n =1

t 2 ∞ (−1)n nπ x nπ t
a π∑
cosh sx + cos sin
33.141. n a a
s sinh as n =1

4 ∞ (−1)n (2n − 1)π x (2n − 1)π t

π∑
cosh sx 1+ cos cos
33.142. 2 n − 1 2 a 2a
s cosh sa n =1

sinh sx xt 2a ∞ (−1)n nπ x nπ t
a π2 ∑
33.143. + 2 sin sin
2
s sinh sa n =1
n a a

sinh sx 8a ∞ (−1)n (2n − 1)π x (2n − 1)π t

π2 ∑
33.144. x+ 2 sin cos
2
s cosh sa n=1
( 2 n − 1) 2 a 2a

cosh sx t 2 2a ∞ (−1)n nπ x ⎛ nπ t ⎞
2a π 2 ∑
33.145. + cos 1 − cos
s 2 sinh sa n =1
n 2
a ⎝ a ⎠

cosh sx 8a ∞ (−1)n (2n − 1)π x (2n − 1)π t

π2 ∑
33.146. t+ 2 cos sin
2
s cosh sa n =1
( 2 n − 1) 2 a 2a

cosh sx 1 2 16a 2 ∞
(−1)n (2n − 1)π x (2n − 1)π t
33.147.
s 3 cosh sa 2
(t + x 2 − a 2 ) − 3
π ∑ (2n − 1)
n=1
3 cos
2a
cos
2a

sinh x s 2π ∞
nπ x
∑ (−1) ne − n π t /a sin
2 2 2
n
33.148. a2 a
sinh a s n =1

cosh x s π ∞
(2n − 1)π x
∑ (−1) n −1
(2n − 1)e − ( 2 n−1) π t / 4 a cos
2 2 2

33.149. a2 2a
cosh a s n =1

sinh x s 2 ∞ (2n − 1)π x

a∑
(−1)n−1 e − ( 2 n−1) π t / 4 a sin
2 2 2

33.150. 2a
s cosh a s n =1

cosh x s 1 2 ∞ nπ x
a a∑
+ (−1)n e − n π t /a cos
2 2 2

33.151. a
s sinh a s n =1

sinh x s x 2 ∞ (−1)n − n π t /a nπ x
+ ∑
2 2 2
33.152. e sin
s sinh a s a π n=1 n a

cosh x s 4 ∞ (−1)n − ( 2 n−1) π t / 4 a (2n − 1)π x

π∑
1+
2 2 2
33.153. e cos
s cosh a s n =1
2 n − 1 2a

sinh x s xt 2a 2 ∞
(−1)n nπ x
+ ∑ (1 − e − n π t /a ) sin
2 2 2
33.154. a π3 3
a
2
s sinh a s n =1
n

cosh x s 1 2 16a 2 ∞
(−1)n (2n − 1)π x
(x − a2 ) + t − 3 ∑ (2n − 1) e − ( 2 n−1) π t / 4 a cos
2 2 2
33.155. 2
s cosh a s 2 π n =1
3
2a
Tabela 3.1 Transformada de Laplace para Varios Dominios em Função do
Tempo
f(t) F(s)
Tabela 3.1 Transformada de Laplace para Varios Dominios em Função do
Tempo
f(t) F(s)
Tabela 3.1 Transformada de Laplace para Varios Dominios em Função do
Tempo

f(t) F(s)