# Applied Mathematics/Laplace Transforms

The Laplace transform is an integral transform which is widely used in physics and engineering. Laplace transform is denoted as $\displaystyle {\mathcal {L}}\left\{f(t)\right\}$ .

The Laplace transform is named after mathematician and astronomer Pierre-Simon Laplace.

## Definition

For a function f(t), using Napier's constant"e" and complex number "s", the Laplace transform F(s) is defined as follow:

$F(s)={\mathcal {L}}\left\{f(t)\right\}(s)=\int _{0}^{\infty }e^{-st}f(t)\,dt$ The parameter s is a complex number:

$s=\sigma +i\omega ,\,$ with real numbers σ and ω.

This $F(s)$ is the Laplace transform of f(t).

## Examples of Laplace transform

Examples of Laplace transform
function result of Laplace transform
$C$ (constant) ${\frac {C}{s}}$ $t$ ${\frac {1}{s^{2}}}$ $t^{n}$ (n is natural number) ${\frac {n!}{s^{n+1}}}$ ${\frac {t^{n-1}}{(n-1)!}}$ ${\frac {1}{s^{n}}}$ $e^{at}$ ${\frac {1}{s-a}}$ $e^{-at}$ ${\frac {1}{s+a}}$ ${\rm {cos}}\ \omega t$ ${\frac {s}{s^{2}+{\omega }^{2}}}$ ${\rm {sin}}\ \omega t$ ${\frac {\omega }{s^{2}+{\omega }^{2}}}$ ${\frac {t^{n-1}}{\Gamma (n)}}$ ${\frac {1}{s^{n}}}$ (n>0)
$\delta (t-a)$ (Delta function) $e^{-as}$ $H(t-a)$ (Heaviside function) ${\frac {e^{-as}}{s}}$ ## Examples of calculation

(1)Suppose $f(t)=C$ (C = constant)
$\int _{0}^{\infty }e^{-st}C\,dt$ $={\frac {C}{s}}$ $=F(s)$ (2)Suppose $f(t)=e^{-at}$ $\int _{0}^{\infty }e^{-st}\cdot e^{-at}\,dt$ $=\int _{0}^{\infty }e^{-(s+a)t}\,dt$ $=\left[{\frac {-e^{-(s+a)t}}{s+a}}\right]_{0}^{\infty }$ $={\frac {1}{s+a}}$ 