Definition of the Laplace Transform of F (t) LAPLACE TRANSFORMS
33 Definition of the Laplace Transform of F (t) LAPLACE TRANSFORMS
33.1. ᏸ{ ( )} Ft =
0 e F t dt st () = fs ∫ ()
In general f (s) will exist for s > a where a is some constant. ᏸ is called the Laplace transform operator.
Definition of the Inverse Laplace Transform of f (s) If ᏸ{F(t)} = f (s), then we say that F (t) = ᏸ –1 {f (s)} is the inverse Laplace transform of f (s). ᏸ –1 is called the
inverse Laplace transform operator.
Complex Inversion Formula The inverse Laplace transform of f (s) can be found directly by methods of complex variable theory. The
result is
2 π i ci −∞ e f s ds () ∫ = 2 π i T lim →∞ c iT e f s ds () ∫ −
where c is chosen so that all the singular points of f (s) lie to the left of the line Re{s} = c in the complex s plane.
LAPLACE TRANSFORMS
Table of General Properties of Laplace Transforms
f (s)
F (t)
33.3. afs 1 () + 2 () bf s
33.5. f (s – a)
33.6. e f (s)
33.7. sf (s) – F (0)
Ft ′ ()
33.8. sfs 2 () − sF () 0 −′ F () 0 F ′′ () t
33.9. sfs n () n
− n s − 1 F () 0 − s − 2 F () 0 − ⋯ − F ( − 1 ′ ) () 0 F (n) (t)
33.10. fs ′ ()
–tF(t)
33.11. f ′′ () s
t 2 F (t)
33.12. f (n) (s)
(–1) n t n F (t)
∫ () 0 F u du
( n − 1 )!
33.15. t
f (s)g(s)
FuGt ()( − u du ∫ ) 0
LAPLACE TRANSFORMS
f (s)
F (t)
33.16. s () ∫ f u du
Ft ()
33.17. − − e sT su () 1 F u du e ∫ 0 F (t) = F(t + T)
πt ∫ 0
e () F u du
33.19. f J
ut F u du
33.20. f t n /2 u − n / + 2 J n ( 2 ut F u du
fs ( + 1 / s )
33.21. 2 J ( 2 ut ( u F u du s )) ( )
u − 32 / e − s 2 / 4 33.22. u () f u du
∫ 0 Γ+ ( u 1 )
∑ k = 1 Q ′ () α k
33.24. k e α k t
Qs ()
P (s) = polynomial of degree less than n,
Q (s) = (s – a 1 )(s – a 2 ) … (s – a n ) where a 1 ,a 2 , …, a n are all distinct.
LAPLACE TRANSFORMS
Table of Special Laplace Transforms
33.30. n n = 123 ,,, ( …
33.33. s 2 + cos at a 2
1 e bt sin at
33.34. ( s − b ) 2 + a 2
e bt cos at
1 sinh at
33.36. s 2 − a 2 a
33.37. s 2 − a 2 cosh at
1 e bt sinh at
33.38. ( s − b ) 2 − a 2
LAPLACE TRANSFORMS
s − b ) 2 2 e cosh − at a
a b − e 33.40. at ( s
33.41. ( s as )( b ) a ≠ b be − ae
1 sin at − at cos at
33.42. ( s 2 + a 22 )
t sin at
33.43. ( s 2 + a 22 )
1 at cosh at − sinh at
33.47. ( s 2 − a 22 )
t sinh at
33.48. ( s 2 − a 22 )
2 a s 2 sinh at at cosh
cosh at + 2 at sinh at
t cosh at
1 ( 3 − at 22 )sin at − 3 at cos at
33.52. ( s 2 + a 23 )
t sin at
− 2 at cos at
33.53. ( s 2 + a 23 )
+ at 22 )sin at − at cos at
( s 2 + a 23 )
8 a 3 s 3 3 t sin at at 2 cos at
( s 2 + a 23 )
LAPLACE TRANSFORMS
f (s)
F (t) s 4 ( 3 − at 22 )sin at + 5 at cos at
33.56. ( s 2 a + 23 )
33.58. ( s 2 a + 23 )
2 a s 3 − 3 as 2 1 2
33.59. ( s 2 a 23 t cos at
s 4 − 6 as 22 + a 4 1 3
33.60. 2 t cos ( at s
+ a 24 )
s 3 − as 2 t 3 sin at
33.61. ( s 2 + 24 a )
24 a
1 ( 3 + at 22 )sinh at 3 at cosh at
( s 2 − 23 a )
at 2 cosh at − t sinh at
33.63. ( s 2 − 23 a )
8 a 3 s 2 at cosh at
+ ( at − 1 )sinh at
( s 2 − 23 a )
8 a 3 s 3 3 t sinh at
8 a s 4 ( 3 + at 22 )sinh at 5 at cosh
8 a s 5 ( 8 + at 22 ) cosh at + 7 at sinh at
33.67. ( s 2 − a 23 )
3 s 2 + a 2 t 2 sinh at
33.68. ( s 2 − 23 a )
2 a s 3 + 3 as 2 1 t 2
33.69. ( 2 cosh s at
− a 23 )
s 4 + 6 as 22 + a 4 1 3
33.70. ( s 2 24 6 t cosh at
s 3 + as 2 t 3 sinh at
33.71. ( s 2 a − 24 )
33.72. s 3 a 3 3 a 2 ⎨
3 sin
cos
e − 3 at / 2
LAPLACE TRANSFORMS
33.75. 3 e at 2
33.78. 4 4 3 (sin at cosh at − cos sinh ) at s at
sin sinh at at
33.79. s 4 4 a + 4
(sin at
cosh at cos sinh ) at at
+ 4 a 4 2a
33.81. 4 4 cos cosh at at
33.82. 4 4 (sinh at sin ) s at
2 a 2 at − cos ) a at
33.83. s 4 4 (cosh
33.84. 4 4 (sinh at + sin ) at
s − a 2a
1 2 (cosh at + cos ) at
33.88. ss ( − a )
− be erfc( bt )
LAPLACE TRANSFORMS
f (s)
F (t)
33.90. 2 2 J at 0 ()
33.91. 2 2 0 () I at
33.92. n
>− n 1 a J at n ()
1 33.93. n >− a I at n ()
s 2 − a 2 bs ( − s e 2 + a 2 )
Jatt )) s 2 0
33.96. ( s 2 + 1 a 232 ) /
33.97. ( s 2 a 232 /
Ft () = nn , ⬉ t <+ n 1 , n = 012… ,,,
See also entry 33.165.
∑ 33.103. r
where [t] = greatest integer ⬉ t
se ( s r ) = − s ( 1 − re − s )
Ft () = rn n , ⬉ t <+ n 1 , n = 012… ,,,
See also entry 33.167.
e −/ as
cos 2 at
LAPLACE TRANSFORMS
e a 4 33.109. t e
erfc / ( a 2 t )
e − as
e ( b bt a + ) erfc ⎛ bt ⎞ +
ln[( s 2 + a 2 ) / a 2 ]
ln[( s aa )]
ln t γ = Euler’s constant = .5772156 …
2(cos at − cos ) bt
ln 2 t γ = Euler’s constant = .5772156 …
γ = Euler’s constant = .5772156 …
γ = Euler’s constant = .5772156 …
LAPLACE TRANSFORMS
erfc ( as / )
π t s 2 a 2 2 a − at 22
e / 4 erfc / ( s 2 a )
e s 2 / 4 a 2 erfc / ( s 2 a )
erf( ) at
e as erfc as
sin as π Si as () + cos as Ci as ()
2 Si as 1 () Ci as () ln ⎛ t + 33.134. a ⎞
a ⎣⎢ 2 ⎦⎥ ⎝⎜ ⎠⎟
0 ᏺ (t) = null function
1 δ (t) = delta function
See also entry 33.163.
LAPLACE TRANSFORMS
f (s)
F (t)
π cos ∑
sinh n sx x 2 ∞ () − 1 nx π
s sinh sa
a a sinh
4 sx ∞ () 1 − n
sin s cosh sa
sin
sin π s sinh as
cosh sx
1 π t s cosh sa
cosh n sx 4 () − 1 ( 2 n − 1 ) π x
cos − )
n = 1 2 n − 1 2 a 2 a sinh sx
a + 2 ∑ n 2 sin π sin n = 1 a a
s 2 sinh sa
sinh sx
2 a 2 a a cosh
s 2 cosh sa
2 a + 2 ∑ 2 1 − cos π ⎞ n = 1 n a a ⎠
s 2 sinh sa
cosh sx
2 2 a 2 a cosh sx
s 2 cosh sa
s 3 cosh sa
sinh xs
− n 2 π () 2 1 ne ta / 2 sin nx π
a 2 sinh − as ∑ n = 1 a
cos cosh
()( 1 n − 1 2 n 1 ) e − ( 2 n − 1 ) 2 2 t / 4 a 2
n = 1 2 a sinh
as
2 xs ∞
a ∑ () − 1 e n sin = 1 2 a
s cosh as
cosh xs
+ ∑ () − 1 e π cos π
− n 2 2 ta / 2 nx
s sinh as
a a n = 1 a sinh xs
s sinh as
a cosh xs
4 ∞ () n
1 − 33.153. 1 e − ( 2 n − 1 )
2 π 2 t / 4 a 2 ( 2 n 1 ) cos
− ππ x
s cosh as
sinh xs
2 33.154. 2 ( 1 e − n π ta / 2 )sin nx π s 2 sinh as
xt 2 a 2 ∞ () 1 − n
a cosh xs
1 2 2 16 a 2 ∞ () 1 n
cos s cosh as
e − ( 2 n − 1 ) 2 π 2 t / 44 a 2
LAPLACE TRANSFORMS
where λ l , λ 2 , … are the positive roots of J 0 ( λ) = 0
1 ∞ e − λ n 2 ta / 2 2 2 J ( xa / ) J ix s 0 ( )
4 ∑ 2 3 n == λ n J
where λ 1 , λ 2 , … are the positive roots of J 0 ( λ) = 0
Triangular wave function
Fig. 33-1 Square wave function
1 as
⎞ s tanh ⎛⎝ 2 ⎠
Fig. 33-2
Rectified sine wave function
coth ⎛ as ⎞
as 22 + π 2 ⎝ 2 ⎠
Fig. 33-3
Half-rectified sine wave function π a
( as 22 − + as π 2 )( 1 − e )
Fig. 33-4 Sawtooth wave function
Fig. 33-5
LAPLACE TRANSFORMS
f (s)
F (t)
Heaviside’s unit function ᐁ(t – a)
e − as
s See also entry 33.138.
Fig. 33-6 Pulse function
e − as ( 1 e − ⑀ − s )
Fig. 33-7 Step function
s ( 1 − e − as ) See also entry 33.102.
Fig. 33-8