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Laplace table, Study notes of Electrical Engineering

University of Engineering and Technology Taxila (UETT)Electrical Engineering

summary of laplace formulae

Typology: Study notes

2013/2014

Uploaded on 12/06/2014

rao.asadali
rao.asadali 🇵🇰

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bg1
Table of Laplace Transforms
(
)
(
)
{
}
1
ftFs
-
=L
(
)
(
)
{
}
Fsft
=L
(
)
(
)
{
}
1
ftFs
-
=L
(
)
(
)
{
}
Fsft
=L
1. 1
1
s
2.
at
1
sa
-
3.
,1,2,3,
n
tn=
K
1
!
n
n
s
+
4.
p
t
, p > -1
(
)
1
1
p
p
s
+
G+
5.
t
3
2
2
s
p
6. 1
2
,1,2,3,
n
tn
-=
K
(
)
1
2
13521
2
n
n
n
s
p
+
××-L
7.
(
)
sin
at
22
a
sa
+
8.
(
)
cos
at
22
s
sa
+
9.
(
)
sin
tat
( )
2
22
2
as
sa
+ 10.
(
)
cos
tat
( )
22
2
22
sa
sa
-
+
11.
(
)
(
)
sincos
atatat
-
( )
3
2
22
2a
sa
+ 12.
(
)
(
)
sincos
atatat
+
( )
2
2
22
2as
sa
+
13.
(
)
(
)
cossin
atatat
-
(
)
( )
22
2
22
ssa
sa
-
+ 14.
(
)
(
)
cossin
atatat
+
(
)
( )
22
2
22
3
ssa
sa
+
+
15.
(
)
sin
atb
+
(
)
(
)
22
sincos
sbab
sa
+
+
16.
(
)
cos
atb
+
(
)
(
)
22
cossin
sbab
sa
-
+
17.
(
)
sinh
at
22
a
sa
-
18.
(
)
cosh
at
22
s
sa
-
19.
(
)
sin
at
bt
e
( )
2
2
b
sab
-+
20.
(
)
cos
at
bt
e
( )
2
2
sa
sab
-
-+
21.
(
)
sinh
at
bt
e
( )
2
2
b
sab
--
22.
(
)
cosh
at
bt
e
( )
2
2
sa
sab
-
--
23.
,1,2,3,
nat
tn=e
K
( )
1
!
n
n
sa
+
- 24.
(
)
fct
1
s
F
cc
æö
ç÷
èø
25.
(
)
(
)
c
ututc
=-
Heaviside Function
cs
s
-
e
26.
(
)
tc
d
-
Dirac Delta Function
cs
-
e
27.
(
)
(
)
c
utftc
-
(
)
cs
Fs
-
e 28.
(
)
(
)
c
utgt
(
)
{
}
cs
gtc
-+eL
29.
(
)
ct
ft
e
(
)
Fsc
-
30.
(
)
,1,2,3,
n
tftn=
K
( )
()
()
1
nn
Fs
-
31.
()
1
ft
t
()
s
Fudu
¥
ò 32.
()
0
t
fvdv
ò
(
)
Fs
s
33.
( ) ()
0
t
ftgd
ttt
-
ò
(
)
(
)
FsGs
34.
(
)
(
)
ftTft
+=
()
0
1
Tst
sT
ftdt
-
-
-
òe
e
35.
(
)
ft
¢
(
)
(
)
0
sFsf- 36.
(
)
ft
¢¢
(
)
(
)
(
)
2
00
sFssff
¢
--
37.
(
)
(
)
n
ft
(
)
(
)
(
)
(
)
(
)
(
)
(
)
21
12
0000
nn
nnn
sFssfsfsff
--
--
¢
----L
pf2

Partial preview of the text

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Table of Laplace Transforms

f ( t ) = L -^1 { F ( s )} F ( s ) = L { f ( t )} f ( t ) = L -^1 { F ( s )} F ( s ) = L{ f ( t )}

s

  1. (^) e at

s - a

  1. t n , n = 1, 2,3,K (^1)

n

n s +^

  1. (^) t p , p > -

1

p

p s +

G +

  1. (^) t (^) 3 2 s^2

p

(^12) t n^ -^ , n = 1, 2,3,K

(^12)

2 n n

n s

p

◊ ◊ L -

7. sin^ ( at^ ) 2 2

a s + a

8. cos^ ( at^ ) 2 2

s s + a

9. t^ sin( at^ )

2 2 2

2 as s + a

10. t^ cos( at^ )

2 2 2 2 2

s a s a

11. sin^ ( at^ ) -^ at^ cos( at )

3 2 2 2

2 a s + a

12. sin^ ( at^ ) +^ at^ cos( at )

2 2 2 2

2 as s + a

13. cos ( at ) - at sin( at )

2 2

2 2 2

s s a

s a

14. cos ( at ) + at sin( at )

2 2

2 2 2

s s 3 a

s a

15. sin ( at + b )

2 2

s sin b a cos b s a

16. cos ( at + b )

2 2

s cos b a sin b s a

17. sinh ( at ) 2 2

a s - a

18. cosh ( at ) 2 2

s s - a

19. e at^ sin( bt )

(^2 )

b s - a + b

20. e at^ cos( bt )

(^2 )

s a s a b

21. e at^^ sinh( bt )

(^2 )

b s - a - b

22. e at^^ cosh( bt )

(^2 )

s a s a b

  1. t n^ e at , n =1, 2,3,K

1

n

n s a

24. f ( ct )

1 s F c c

Ê ˆ

Á ˜

Ë ¯

25. uc^ (^ t^ )^ =^ u t (^^ - c )

Heaviside Function

cs

s

e^ -

26.^ d^ (^ t^ - c )

Dirac Delta Function

e -^ cs

27. uc^ ( t^ ) f^ ( t^ -^ c )^ e^ - cs^ F^ ( s ) 28. uc^ ( t^ ) g t (^ ) e -^^ cs^ L{ g t (^ + c )}

29. e ct^^ f^ ( t ) F^ ( s^ -^ c ) 30.^ t^ nf^ ( t^ ) ,^ n^ =^ 1, 2,3,K^ ( 1 ) (^ )^ ( )

n (^) n

  • F s

f t t

s F u du (^ )

Ú 32.^0 (^ )

t

Ú^ f^ v dv (^ )

F s s

t

Ú^ f^ t^ -^ t^ g^ t^ dt F^ (^ s G s )^ (^^ ) 34.^ f^ (^ t^ +^ T^ )^ =^ f^ (^ t )^0 ( )

T (^) st

sT

  • f t dt

Ú e

e

35. f^ ¢^ ( t ) sF^ ( s^ ) -^ f ( 0 ) 36. f^ ¢¢^ ( t ) s F^2^ ( s^ ) -^ sf^ ( 0 )^ - f^ ¢( 0 )

37. f (^ n )^ ( t ) s Fn^ ( s ) - s n^ -^1 f ( 0 ) - s n -^2 f ¢( 0 ) L- sf (^ n^ -^2 )^ ( 0 ) - f (^ n -^1 )^ ( 0 )

Table Notes

  1. This list is not a complete listing of Laplace transforms and only contains some of the more commonly used Laplace transforms and formulas.
  2. Recall the definition of hyperbolic functions.

cosh ( ) sinh( )

t t t t t t

+ -^ - -

e e e e

  1. Be careful when using “normal” trig function vs. hyperbolic functions. The only difference in the formulas is the “+ a^2 ” for the “normal” trig functions becomes a “- a^2 ” for the hyperbolic functions!
  2. Formula #4 uses the Gamma function which is defined as

( )^ t^ 0^ x^ xt^^1 dx

  • (^) - - G = (^) Ú e

If n is a positive integer then,

G ( n + 1 ) = n!

The Gamma function is an extension of the normal factorial function. Here are a couple of quick facts for the Gamma function

p p p p n p p p p n p

p

G + = G

G +

G

Ê ˆ

G Á ˜=

Ë ¯

L