TABLICA LAPLACE-ovih TRANSFORMACIJA

Lik ( )F p Original ( )f t

1. 0

( )pte f t d∞

−∫ t ( )f t

2. ( )F p 1

( )2

c jpt

c j

e F p dpj

+ ∞

− ∞π ∫

3. ( )CF p ( )Cf t 4. 1 2( ) ( )F p F p+ 1 2( ) ( )f t f t+

5. ( ) (0)pF p f− ( )df tdt

6. 2 '( ) (0) (0)p F p pf f− − 2

2( )d f t

dt

7. ( 1)

1

( ) (0)n

n n i i

i

p F p p f− −

=−∑ ( )n

nd f tdt

8. ( )F pp

0

( )t

f t dt∫

9. ( )nF pp

0 0 0

( )t t t

dt dt f t dt∫ ∫ ∫…

10. ( )n

nd F pdp

( 1) ( )n nt f t−

11. 1 2( ) ( )F p F p 1 20

( ) ( )t

f f t dτ − τ τ∫

12. ( )p

F p dp∞

∫ ( )f tt

13. ( )pTe F p− ( )f t T−

14. ( )F p a+ ( )ate f t−

15. ( , )F p aa

∂∂

( , )f t aa

∂∂

16. 1

, 0( )

0, 0

tt

t

∞ =⎧⎪δ = ⎨ ≠⎪⎩

17. 1p

1, 0( )

0, 0

th t

t

>⎧⎪= ⎨ <⎪⎩

18. Cp

0C t >

19. 1!nn

p + 0,1,2,nt n = …

1

20. 1p a±

Re Reate p >∓ a

21. 1

p j± ω j te ω∓

22. ( )m

A p a− atm

eA

23. 21

p t

24. 1( )p p a+

1(1 )atea

−−

25. 1

( )(p a p b+ + )

at bte eb a

− −−−

26. 21

( )p a+ atte−

27. 2( )

p

p a+ (1 ) atat e−−

28. 1!

( )nn

p a ++ 0,1,2,n att e n− = …

29. ( )mp nAp p a

+−

( ) atn m n

eAa A Aa

− + +

30. 2 2p

ω+ ω

sin tω

31. 2 2sin cosp

p

± ψ + ω+ ω

ψ sin( )tω ± ψ

32. 2 2p

p + ω cos tω

33. 2 2cos sinp

p

ψ ω ψ+ ω∓ cos( )tω ± ψ

34. 2 2( )p a

ω+ + ω

sinate t− ω

35. 2 2( )

p a

p a

++ + ω

cosate t− ω

36. 2 2p

ω− ω

sinh tω

37. 2 2p

p − ω cosh tω

38. 2 2( )p a

ω+ − ω

sinhate t− ω

39. 2 2( )

p a

p a

++ − ω

coshate t− ω

40. ( )(mp n

A p a p b+

+ + ) ( ) ( )

( )

at btn am e n bm eA b a

− −− − −−

41. 2( )

p n

p a

++

[ ]( ) 1 atn a t e−− +

2

42. 2

2 4

mp n

Ap Bp C

B AC

++ +

>

0 00

20

cosh sinh

2

at m n amt te A A

B Ca a

A A

− −⎡ ⎤ω + ω⎢ ⎥ω⎣ ⎦

= ω = −

43. 2

2 4

mp n

Ap Bp C

B AC

++ +

=

2

at m n amte

A AB

aA

− −⎡ ⎤+⎢ ⎥⎣ ⎦

=

44. 2

2 4

mp n

Ap Bp C

B AC

++ +

<

0 00

20

cos sin

2

at m n amt te A A

B Ca a

A A

− −⎡ ⎤ω + ω⎢ ⎥ω⎣ ⎦

= ω = −

45. 31

p

2

2 !t

46. 21

( )p p a+

21ate at

a

− + −

47. 21

( )p p a+

21 (1 ) atat e

a

−− +

48. 31

( )p a+

2

2

att e−

49. 3( )

p

p a+ 2(2 )

2

att at e−−

50. 2

3( )

p

p a+

2 2(2 4 )2

atat a t e−− +

51. ( )(p c

p p a p b+

+ + )

( ) ( )at btc c a c be e

ab a a b b b a− −− −

+ +− −

52. 21

( ) (p a p b+ + ) [ ]

2( ) 1

( )

bt ate b a t e

b a

− −+ − −

−

53. 1

( )( )(p a p b p c+ + + ) ( ) ( ) ( )

( )( )( )

at bt ctc b e a c e b a ea b b c c a

− − −− + − + −− − −

54. 2

( )( )(mp np q

p a p b p c+ +

+ + + )

2 2 2

( )( ) ( )( ) ( )( )at bt ctma na q mb nb q mc nc qe e

a b a c b a b c c a c be− − −− + − + − +

+ +− − − − − −

55.

2

2

2

( )

4

mp np q

p Ap Bp C

B AC

+ ++ +

>

( )

0 00 0

20

cosh sinh

2

at n am aqm qqe t

A CA CC

B Ca a

A A

− −t

⎡ ⎤⎛ ⎞−−+ ω + ⎜ ⎟ ω⎢ ⎥ω ω⎝ ⎠⎣ ⎦

= ω = −

56.

2

2

2

( )

4

mp np q

p Ap Bp C

B AC

+ ++ +

=

( ) ( )

2

at m q n am aqqe t

A C A CCB

aA

− −⎡ ⎤− −+ +⎢ ⎥⎣ ⎦

=

3

57.

2

2

2

( )

4

mp np q

p Ap Bp C

B AC

+ ++ +

<

( )

0 00 0

20

cos sin

2

at n am aqm qqe t

A CA CC

B Ca a

A A

− t−⎡ ⎤⎛ ⎞−−+ ω + ⎜ ⎟ ω⎢ ⎥ω ω⎝ ⎠⎣ ⎦

= ω = −

58. 31

( )p p a+

2 2

31 (1 0,5 ) atat a t e

a

−− + +

59. 2 21

( )p + ω 2 3

sin cos

2

t t tω − ω ωω

60. 2 2 21

( )p p + ω

3sint tω − ωω

61. 2 2( )

p

p + ω 2 sin2t tω

ω

62. 4 4p

p − ω

2cosh cos

2

t tω − ωω

63. 2 2 2 21

( )(p a p b− − ) 2 2

sinh sinh

( )

b at a b

ab a b

t−−

64. 2 2 2 21

( )(p a p b+ + ) 2 2

sin sin

( )

b at a bt

ab b a

−−

4