Introduction : In this page we write a list of Laplace transform formulas of basic functions and those are commonly used in all the applications of science and engineering. Using these formulas we can obtain Laplace transform of any function which is written as addition or multiplication or division of these basic functions.
Laplace transform
Let f(t) be a function which must be defined for all non-negative real numbers then Laplace transform of f(t) is denoted as F(s) where s is a parameter from set of all complex numbers and obtained using formula below
\[{\cal L}\left( {f(t)} \right)=F(s) = \int\limits_0^\infty f (t){e^{ – st}} dt\]
Laplace transform of algebraic functions
- \({\cal L}\left( {0} \right)= 0\)
- \({\cal L}\left( {1} \right)=\frac{1}{s} \)
- \({\cal L}\left( {k} \right)=\frac{k}{s} \) where k is any constant
- \({\cal L}\left( {t} \right)=\frac{1}{s^2} \)
- \({\cal L}\left( {t^k} \right)=\frac{k!}{s^{k+1}} \) for any natural number k
- \({\cal L}\left( {t^{-\frac{1}{2}}} \right)=\frac{{\sqrt \pi }}{{{s^{\frac{1}{2}}}}} \)
- \({\cal L}\left( {t^{k} \right)=\frac{{\Gamma (k + 1)}}{{{s^{k + 1}}}}\ ) where k>-1 is a fraction number.
Laplace transform of exponential functions
- \({\cal L}\left( {e^{kt}} \right)= \frac{1}{s-k}\)
- \({\cal L}\left( {e^{-kt}} \right)=\frac{1}{s+k} \)
- \({\cal L}\left( {te^{kt}} \right)= \frac{1}{(s-k)^2}\)
- \({\cal L}\left( {te^{-kt}} \right)= \frac{1}{(s+k)^2}\)
- \({\cal L}\left( {p^t} \right) =\frac{1}{s-\log (p)}\) where p>0
Laplace transform of trigonometric functions
- \({\cal L}\left( {sin(t)} \right)=\frac{1}{s^2+1}\)
- \({\cal L}\left( {cos(t)} \right)=\frac{s}{s^2+1}\)
- \({\cal L}\left( {sin(pt)} \right)=\frac{1}{s^2+p^2}\)
- \({\cal L}\left( {cos(pt)} \right)=\frac{s}{s^2+p^2}\)
- \({\cal L}\left( {e^{kt}sin(pt)} \right)=\frac{p}{(s-k)^2+p^2} \)
- \({\cal L}\left( {e^{-kt}sin(pt)} \right)= \frac{p}{(s+k)^2+p^2}\)
- \({\cal L}\left( {e^{kt}cos(pt)} \right)= \frac{s-k}{(s-k)^2+p^2}\)
- \({\cal L}\left( {e^{-kt}cos(pt)} \right)= \frac{s+k}{(s+k)^2+p^2}\)