Laplace Transform#
Definition#
Properties#
\(\displaystyle \Laplace{f+g} = \Laplace{f} + \Laplace{g}\).
\(\displaystyle \Laplace{cf} = c\Laplace{f}\) for any constant \(c\).
\(\displaystyle \Laplace{e^{at}f(t)}(s)=F(s-a)\).
\(\displaystyle \Laplace{f'}(s)=sF(s) - f(0)\).
\(\displaystyle \Laplace{f''}(s)=s^2F(s) -sf(0)- f'(0)\).
Table of Laplace Transforms#
\(f(t)\) |
\(F(s)=\Laplace{f}(s)\) |
|---|---|
\(1\) |
\(\dfrac{1}{s}\), \(\quad s>0\) |
\(e^{at}\) |
\(\dfrac{1}{s-a}\), \(\quad s>a\) |
\(t^n\) |
\(\dfrac{n!}{s^{n+1}}\), \(\quad s>0\) , \(\quad n=1,2,\ldots\) |
\(\sin bt\) |
\(\dfrac{b}{s^2+b^2}\), \(\quad s>0\) |
\(\cos bt\) |
\(\dfrac{s}{s^2+b^2}\), \(\quad s>0\) |
\(e^{at} t^n\) |
\(\dfrac{n!}{(s-a)^{n+1}}\), \(\quad s>a\), \(\quad n=1,2,\ldots\) |
\(e^{at} \sin bt\) |
\(\dfrac{b}{(s-a)^2+b^2}\), \(\quad s>a\) |
\(e^{at} \cos bt\) |
\(\dfrac{s-a}{(s-a)^2+b^2}\), \(\quad s>a\) |
\(\sinh bt\) |
\(\dfrac{b}{s^2-b^2}\) |
\(\cosh bt\) |
\(\dfrac{s}{s^2-b^2}\) |