FORMULA FOR LAPLACE TRANSFORM and its inverses

S.NO f(t) L[f(t)] L-1 (L[f(t)])= f(t)

1 1 L[1] = 1/s 1/s =1

2 t L[t] = 1/s2 1/s2 = t

3 tn L[tn] = n!/sn+1 1/sn = tn-1/(n-1)!

4 tn L[tn] = Ғ(n+1)/sn+1,n-non integer

5. eat L[eat] = 1/s-a 1/s-a = eat

6 e−at L[e−at] = 1/s+a 1/s+a = e−at

7 Sinat L[Sinat] = a/s2+a2 a/s2+a2=sinat ,

1/s2+a2 = sinat/a

8 Cosat L[Cosat] == s/s2+a2 s/s2+a2 = Cosat

9 Sinhat L[Sinhat] == a/s2−a2 a/s2−a2 = sinhat

10 Coshat L[Coshat] == s/s2−a2 s/s2−a2 = coshat

Properties of LAPLACE TRANSFORM

S.NO

PROPERTY f(t) L[f(t)]

1 scale f(at) 1/a F(s/a)

2 derivative f '(t) sL[f(t)]-f(0)

f ' ' (t) s2 L [ f (t ) ]−sf (0 )−f '(0)

3 Division by t 1/t f(t)∫s

∞

L [ f ( t ) ] ds

4 Multiple by t tn f(t) (-1)n dn/dsn L[f(t)]

5 Initial value theorem

limt →0

f ( t) lims→∞

sF (s )

6 Final value theorem

limt →∞

f ( t) lim s→0sF (s)

7 First shifting theorem

eat f (t) F[s-a]

e−at f (t) F[s+a]

Properties of inverses LAPLACE TRANSFORM

S.NO

PROPERTY

1 First shifting L-1[F(s+a)] e−at L−1[F(s)]

2 derivatives L-1[F '(s)] (identification : :

s+any term¿¿

-t L-1 F(s)

3 Division by sL-1[ F (s)

s ] (identification : :

any terms (ont term )

∫0

t

L−1[F ( s )] dt

4 Multiple by s L-1[s F(s)] (identification : :

squadratic eq . ,

d/dt L-1F(s)

5 L-1[Log, cot ,tan functions]-1/t L-1

dds

[F(s)]

6 Convolution theorem

L[(f*g)] = L[f(t)]L[g(t)]