pdf tabel laplace compress

  

© 2004 Paul Dawkins

© 2004 Paul Dawkins 11

Table of Laplace Transforms Table of Laplace Transforms

(

( ) )

¹¹

{ { ( ( )) }}

 f

 f t t

= =

LL^-- F F ss     ^{F  F s}

( ( )

^s

) = =

^LL

{ {

^{f f t} 

( ( ))

^t 

}}

    ^{f  f t}

( ( )

^t

) = =

^LL--11

{ {

^{F F ss}

( ( )) }}

    ^{F  F s}

( ( )

^s

) = =

^LL

{ {

^{f f t} 

( ( ))

^t 

}}

  

1. 1

1. 1 11

 s

 s    2.2. ee^at at    11

 s  s a

- -

a   3

3. . t t ⁿⁿ,, nn

 =  =

11,, 22,,3   3,,^KK  

1 1

!!

n n

n n  s

 s   ⁺⁺    4.4. t t  ,, p > -1 p > -1

( ( ))

1 1

1 1

 p  p

 p  p  s  s   ⁺⁺

G G + +

  

5.

5. t t    ³³₂₂

2 2 s s

p  p 

   6.6.

1 1 2

2,, 11,, 22,,33,,

n

t n

t nn

- -

=

=

   ^KK  

( (

¹¹²²

))

1

1 33 55 22 11 2

2^{nn   nn} n n  s  s

p  p  +

+

× × × ×

^LL

- -

   7.

7. ^sinsin

( ( ))

   ^atat ₂₂ ^aa ₂₂

 s

 s

+ +

aa    ^{8.8. coscos}

( ( ))

   ^at at     ₂₂ ^s s ₂₂

 s

 s

+ +

aa    9.

9. ^{t t sinsin}

( ( ))

^at at    

( (

^{22 22}

))

²²

2 2asas  s

 s

+ +

aa    ^{10.10. t t coscos}

( ( ))

^at at    

( ( ))

2 2 22

2 2 2

2 22

 s  s aa  s  s aa

- - +

+

   11.

11. ^ssiinn

( ( )

^{aat t a}

) - -

^{at t ccooss}

( ( ))

^{aat  t}    

( ( ))

3 3

2 2 2

2 22

2 2aa  s

 s

+ +

aa    ^{12.12. ssiinn}

( ( )

^{aat t a}

) + +

^{at t ccooss}

( ( ))

^{aat  t}    

( ( ))

2 2

2 2 2

2 22

2 2asas  s

 s

+ +

aa    13.

13. ^ccooss

( ( )

^{aat t a}

) - -

^{at t ssiinn}

( ( ))

^{aat  t}    

( ( ))

( ( ))

2 2 22

2 2 2

2 22

 s

 s s s aa  s  s aa

- - +

+

   ^{14.14. ccooss}

( ( )

^{aat t a}

) + +

^{at t ssiinn}

( ( ))

^{aat  t}    

( ( ))

( ( ))

2

2 22

2 2 2

2 22

3 3  s

 s s s aa  s

 s aa

+ + +

+

  

15.

15. ^sinsin

( (

   ^{at at b}

+ +

^b

))

  

( ( ) ) ( ( ))

2 2 22

ssiinn ccooss  s

 s b b a a bb  s

 s aa

+ + +

+

   ^{16.16. coscos}

( (

   ^{at at b}

+ +

^b

))

  

( ( ) ) ( ( ))

2 2 22

ccooss ssiinn  s

 s b b a a bb  s

 s aa

- - +

+

  

17.

17. ^sinhsinh

( ( ))

   ^at at     ₂₂ ^aa ₂₂

 s

 s

- -

aa    ^{18.18. coshcosh}

( ( ))

   ^at at     ₂₂ ^s s ₂₂

 s

 s

- -

aa    19.

19. ^{eeat at sinsin}

( ( ))

^bt bt    

( ( ))

^{22 22}

b b  s

 s a

- -

a

+ +

bb    ^{20.20. eeat at coscos}

( ( ))

^bt bt    

( ( ))

^{22 22}

 s  s aa  s

 s a a bb

- - -

- + +

   21.

21. ^{eeat at sinhsinh}

( ( ))

^bt bt    

( ( ))

^{22 22}

b b  s

 s a

- -

a

- -

bb    ^{22.22. eeat at coshcosh}

( ( ))

^bt bt    

( ( ))

^{22 22}

 s  s aa  s

 s a a bb

- - -

- - -

   2

233. . t t ^{n n at} _ee^at ,, nn

 =  =

   11,, 22,, 33,,^KK  

( ( ))

¹¹

!!

n n

n n  s

 s a

- -

a    ^{++    24.24.  f  f ct}

( ( ))

^{ct 11 F  F    ss}

c c cc

æ æ öö ç ç ÷÷

è è øø

  

25.

25. u t u t u cc

( ( ) ) ( = =

u t c

( ))

t c

- -

  

Heaviside Function Heaviside Function  

cs cs

 s  s

-

ee-

   26.26. ^d d 

( (    -

^{t t cc}

- ))

  

Dirac Delta Function Dirac Delta Function  

cs -cs

ee-    27.

27. u t f u t f t cc

( ( ) ) ( ( ))

t cc

- -

   ^{ee--cscs F  F ss}

( ( ))

   28.28. u u t cc

( ( )

t g

) (

g t  

( ))

t     ^{ee--cscs   LL}

{ { ( (

^{t t cc}

+ + )) }}

  

29.

29. ^{eect ct  f  f t} 

( ( ))

^{t      F  F s}

( (

^{s cc}

- - ))

   30.30. ^{t t f nn f t}

( ( ))

^{t ,, nn}

 =  =

^{11,, 22,,3   3,,KK}  

( ( )) - -

^{11   nn F  F    ( ( ))   nn}

( ( ))

^ss   

31.

31. ^{11 f  f t} 

( ( ))

^t 

t 

t    

( ( ))

 s

 s  F  F u u dudu

¥

òò

¥    ^32.32.

òò

₀₀^{t t  f  f v}

( ( ))

^{v dvdv    F  F ss}

( ( ))

 s  s    33.

33.

( ( ) ) ( ( ))

0 0 t 

t  f  f t t

- -

^{t t} g g ^{t t} d d ^t t 

òò

    ^{F  F s}

( ( )

^{s G}

) (

^{G ss}

( ))

   ^{34.34.  f  f t}

( ( )

^{t T}

+ +

^T

) ( = =

^{f f t} 

( ))

^{t     00}

( ( ))

1 1

T  T   st  st 

 sT   sT 

 f

 f t t dt dt 

- -

-

-

-

-

òò

^ee

ee   

35.

35.  ^{f  f t} 

¢¢ ( ( ))

^{t      sF  sF s}

( ( )

^s

) ( - -

^f f 

( ))

⁰⁰    36.36.  ^{f  f t} 

¢¢¢¢ ( ( ))

^{t      s  s F 22F s}

( ( )

^s

) - -

^{sf sf}

( ( )

⁰⁰

) - -

^f f 

 ¢ ¢ ( ( ))

⁰⁰   

37.

37.  ^{f  f ( ( ))nn}

( ( ))

^{t t      s  s F n n F s}

( ( )

^s

) - -

^{s s f n n - - 11f}

( ( )

⁰⁰

) - -

^{s s nn--22f f}

¢¢ ( ( ))

^{00 LL}

- -

   ^{sf sf ( ( n n - - 22))}

( ( ))

⁰⁰

- -

  ^{f f ( ( nn--11))}

( ( ))

⁰⁰   

  

© 2004 Paul Dawkins

© 2004 Paul Dawkins 22

Table Notes Table Notes

1.

1.   This list is not inclusive and only contains soThis list is not inclusive and only contains some of the more commonly usedme of the more commonly used Laplace transforms and

Laplace transforms and formulas.formulas.

2.

2.   Recall the definition of hyperbolic trig functRecall the definition of hyperbolic trig functions.ions.

(

( ) ) ( ( ))

ccoosshh ssiinnhh

2

2 22

t

t tt tt tt   

t

t t t 

-

- --

+

+ - -

=

=

^{ee ee}

= =

^{ee ee}   

3.

3.   Be careful when usinBe careful when using “normal” trig function vs. hyperbolic trig “normal” trig function vs. hyperbolic trig functions. g functions. TheThe only difference in the formulas is the “+ a

only difference in the formulas is the “+ a²²” for the “normal” trig funct” for the “normal” trig functionsions  becomes a “- a

 becomes a “- a²²” for the hyperbolic trig functions!” for the hyperbolic trig functions!

4.

4.   Formula #4 uses the Gamma Formula #4 uses the Gamma function which is defined asfunction which is defined as

( ( ))

¹¹

0 0

 x  x t t 

t

t ^{¥¥ - -} x x d^-- dxx

G

G = = òò ee

If

Ifnn is a po is a positive integer then,sitive integer then,

( (

^{n n 11}

))

^nn!!

G

G + + = =

  

The Gamma function is

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

couple of quick facts for the Gamma functionhe Gamma function

(

( ) ) ( ( )) ( )

( ) ( ( ) ) ( ( ))    ( ( )) ( ( ))

1 1 1

1 22 11

1 1 2 2  p

 p p p pp

 p  p nn  p

 p p p p p p p nn

 p  p

p  p 

G

G + = + = G G

G G + + +

+ + + + + - - = = G G æ

æ öö G

G ç ç ÷÷ = = è è øø

L

L