Formulae of Laplace Transformation | laplace transformation | Laplace Transform Table | Properties of Laplace Transformation

The method of Laplace Transformation has the advantage of directly giving the solution of differential equations with given boundary values without the necessity of first finding the general solution and then evaluating from it the arbitrary constant.
Laplace Transformation of function f(t) is

L[f(t)]=\int_{0}^{\infty }e^{-st}f(t)dt

Some Laplace Formulae are given below

  1. L[1]=\frac{1}{s}
  2. L[t^{n}]=\frac{n!}{s^{n+1}} if n = 0, 1, 2, 3…..

otherwise

L[t^{n}]=\frac{\Gamma (n+1)}{s^{n+1}}

3. L[e^{at}]=\frac{1}{s-a}

4. L[sinat]=\frac{a}{s^{2}+a^{2}}

5. L[cosat]=\frac{s}{s^{2}+a^{2}}

6. L[sinhat]=\frac{a}{s^{2}-a^{2}}

7. L[coshat]=\frac{s}{s^{2}-a^{2}}

8. L[e^{at}f(t)]=f(s-a)\;\;\;\; (First Shift Property)

e.g. L[e^{at}t^{n}]=\frac{n!}{(s-a)^{n+1}}

9. Multiplication of  tn Property

if  L[f(t)] = f(s)
    then

\;\;\;\;L[t^{n}f(t)]=(-1)^{n}\frac{d^{n}}{ds^{n}}f(s)

e.g. Find the Laplace Transformation of tcos\alpha t

\;\;\;\;L[cos\alpha t]=\frac{s}{s^{2}+\alpha ^{2}}

then

\;\;\;\;L[tcos\alpha t]= (-1)\frac{d}{ds}(\frac{s}{s^{2}+\alpha ^{2}})\;\;\;\; \because n = 1

\;\;\;\;L[tcos\alpha t] = \frac{s^{2}-\alpha ^{2}}{(s^{2}+\alpha ^{2})^{2}}

10. Division by “t” property

if L[f(t)] = f(s)

  then 

\;\;\;\;L[\frac{f(t)}{t}] = \int_{s}^{\infty }f(s)ds

e.g. Find the Laplace Transformation of  \frac{sint}{t}

\;\;\;\;L[sint] = \frac{1}{s^{2}+ 1}

\;\;\;\;L[\frac{sint}{t}] = \int_{s}^{\infty }\frac{1}{s^{2}+1} = cot^{-1}s

11. Change of Scale Property
    if L[f(t)] = f(s)

  then 

\;\;\;\;L[f(at)] = \frac{1}{a}f(\frac{s}{a})

12. Second Shift Property or Heaviside’s Shifting Theorem 
    if L[f(t)] = f(s)

\;\;\;\;L[G(t)]=e^{-as}L[f(t)] =e^{-as}f(s)

13. Laplace Transformation of Periodic function

L[f(t)]=\frac{1}{1-e^{-sT}}\int_{0}^{T}e^{-st}f(t)dt

where “T” is Period of the Function.

14. Laplace Transformation of Bessel’s function

J_{0}(x)=1-\frac{x^{2}}{2^{2}}+\frac{x^{4}}{2^{2}.4^{2}}-\frac{x^{6}}{2^{2}.4^{2}.6^{2}}+....\\\\L[J_{0}(x)]=\frac{1}{\sqrt{s^{2}+1}}\\\\J_{0}^{'}x = -J_{1}x\\\\L[J_{1}x]= -L[J_{0}^{'}x]=-[sL[J_{0}x]-1]\\\\L[J_{1}x]= 1-\frac{s}{\sqrt{s^{2}+1}}

15. Laplace Transformation of  Error Function

\\erf\sqrt{x}=\frac{2}{\pi}\int_{0}^{\sqrt{x}}e^{-t^{2}}dt\\\ L[erf\sqrt{x}]=\frac{1}{s\sqrt{s+1}}

16. Laplace Transformation of integrals 

if f[t] = f[s]

then

\;\;\;\;L{ \int_{0}^{t}f[t]dt}=\frac{1}{s}f(s)

17. Laplace Transformation of Derivatives

if f(t) = f(s) then

\;\;\;\;L[f^{'}(t)]=sL[f(t)]-f(0)

where f^{'}(t) is derivative of f(t)

Author: AJAZ UL HAQ

AJAZ UL HAQ has above 8 years of Experience in Electrical Power Transmission, Distribution and Substation. Presently He is working with KEI Industries Limited as Engineer-EPC/EHV.

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