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


6                L[        ] = 1/s+a      1/s+a =


7        Sinat   L[Sinat] = a/            a/                    ,

                                          1/         = sinat/a


8        Cosat   L[Cosat] == s/           s/         = Cosat


9       Sinhat   L[Sinhat] == a/          a/         = sinhat



10      Coshat   L[Coshat] == s/          s/         = coshat
Properties of LAPLACE TRANSFORM

S.NO   PROPERTY                    f(t)         L[f(t)]


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


2      derivative         (t)                sL[f(t)]-f(0)




                                                   (0)


3      Division by t    1/t f(t)




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


5      Initial value
       theorem


6      Final value
       theorem


7      First shifting                           F[s-a]
       theorem

                                                F[s+a]
Properties of inverses LAPLACE TRANSFORM

S.NO                                   PROPERTY


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


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




3       Division by s    L-1       (identification :

                         :




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




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


6       Convolution                          L[(f*g)] = L[f(t)]L[g(t)]
        theorem

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Unit v laplace transform(formula)

  • 1. 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. L[ ] = 1/s-a 1/s-a = 6 L[ ] = 1/s+a 1/s+a = 7 Sinat L[Sinat] = a/ a/ , 1/ = sinat/a 8 Cosat L[Cosat] == s/ s/ = Cosat 9 Sinhat L[Sinhat] == a/ a/ = sinhat 10 Coshat L[Coshat] == s/ s/ = coshat
  • 2. Properties of LAPLACE TRANSFORM S.NO PROPERTY f(t) L[f(t)] 1 scale f(at) 1/a F(s/a) 2 derivative (t) sL[f(t)]-f(0) (0) 3 Division by t 1/t f(t) 4 Multiple by t tn f(t) (-1)n dn/dsn L[f(t)] 5 Initial value theorem 6 Final value theorem 7 First shifting F[s-a] theorem F[s+a]
  • 3. Properties of inverses LAPLACE TRANSFORM S.NO PROPERTY 1 First shifting L-1[F(s+a)] [F(s)] 2 derivatives L-1[ ] (identification : -t L-1 F(s) : 3 Division by s L-1 (identification : : 4 Multiple by s L-1[s F(s)] (identification : d/dt L-1F(s) : 5 L-1[Log, cot ,tan functions] -1/t L-1 6 Convolution L[(f*g)] = L[f(t)]L[g(t)] theorem