# Common Laplace Transform Pairs

## Laplace Transform

The linear time invariant (LTI) system is described by differential equations. The Laplace transform is a mathematical tool which converts the differential equations in time domain into algebraic equations in the frequency domain (or s-domain).

If $\mathrm{\mathit{x\left ( t \right )}}$ is a time function, then the Laplace transform of the function is defined as −

$$\mathrm{\mathit{L\left [ x\left ( t \right ) \right ]\mathrm{=}X\left ( s \right )\mathrm{=}\int_{-\infty }^{\infty }x\left ( t \right )e^{-st}\:dt\; \; \cdot \cdot \cdot\left ( \mathrm{1} \right ) }}$$

Where, s is a complex variable and it is given by,

$$\mathrm{\mathit{s\mathrm{=}\sigma \mathrm{+ }j\omega }}$$

## Inverse Laplace Transform

The inverse Laplace transform is defined as −

$$\mathrm{\mathit{L^{\mathrm{-1}}\left [X\left ( s \right ) \right ]\mathrm{=}x\left ( t \right )\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}\pi j}\int_{\sigma -j\infty }^{\sigma \mathrm{+ }j\infty }X\left ( s \right )e^{st}\:ds\; \; \cdot \cdot \cdot\left ( \mathrm{2} \right ) }}$$

The equations (1) and (2) constitute the Laplace transform pair, and it may be represented as,

$$\mathrm{\mathit{x\left ( t \right )\overset{LT}{\leftrightarrow} X\left ( s \right ) }}$$

## Common Laplace Transform Pairs

Following table provides a number of Laplace transforms. The table also specifies the region of convergence (ROC) −

Function
$\mathrm{\mathit{\left\{ x\left ( t \right )\mathrm{=}L^{-1}\left [ x\left ( t \right ) \right ]\right\}}}$
Laplace Transform
$\mathrm{\mathit{\left\{ L\left [ x\left ( t \right ) \right ]\mathrm{=}X\left ( s \right )\right\}}}$
Region of
Convergence (ROC)
$\mathrm{\mathit{\delta \left ( t \right )}}$1$\mathit{\mathrm{All}\; s}$
$\mathrm{\mathit{\delta \left ( t-a \right )}}$$\mathrm{\mathit {e^{-as}}}$$\mathit{\mathrm{All}\; s}$
$\mathrm{\mathit{u \left ( t \right )}}$$\mathrm{\mathit {\frac{\mathrm{1}}{s}}}$$\mathit{\mathrm{Re}\left ( s \right )> \mathrm{0}}$
$\mathrm{\mathit{u \left ( t-a \right )}}$$\mathrm{\mathit {\frac{e^{-as}}{s}}}$$\mathit{\mathrm{Re}\left ( s \right )> \mathrm{0}}$
$\mathrm{\mathit{u \left ( -t \right )}}$$\mathrm{\mathit -{\frac{\mathrm{1}}{s}}}$$\mathit{\mathrm{Re}\left ( s \right )< \mathrm{0}}$
$\mathrm{\mathit{tu \left ( t \right )}}$$\mathrm{\mathit {\frac{\mathrm{1}}{s^{\mathrm{2}}}}}$$\mathit{\mathrm{Re}\left ( s \right )> \mathrm{0}}$
$\mathrm{\mathit{t^{\mathrm{2}}u \left ( t \right )}}$$\mathrm{\mathit {\frac{\mathrm{2!}}{s^{\mathrm{3}}}}}$$\mathit{\mathrm{Re}\left ( s \right )> \mathrm{0}}$
$\mathrm{\mathit{t^{n}u \left ( t \right )}}$$\mathrm{\mathit {\frac{n!}{s^{\left ( n\mathrm{+ }\mathrm{1} \right )}}}}$$\mathit{\mathrm{Re}\left ( s \right )> \mathrm{0}}$
$\mathrm{\mathit{e^{-at}u \left ( t \right )}}$$\mathrm{\mathit {\frac{\mathrm{1}}{\left ( s\mathrm{+ }a \right )}}}$$\mathit{\mathrm{Re}\left ( s \right )> -a}$
$\mathrm{\mathit{e^{at}u \left ( t \right )}}$$\mathrm{\mathit {\frac{\mathrm{1}}{\left ( s-a \right )}}}$$\mathit{\mathrm{Re}\left ( s \right )> a}$
$\mathrm{\mathit{te^{-at}u \left ( t \right )}}$$\mathrm{\mathit {\frac{\mathrm{1}}{\left ( s\mathrm{+ }a \right )^{\mathrm{2}}}}}$$\mathit{\mathrm{Re}\left ( s \right )> -a}$
$\mathrm{\mathit{t^{n}e^{-at}u \left ( t \right )}}$$\mathrm{\mathit {\frac{n!}{\left ( s\mathrm{+ }a \right )^{n\mathrm{+ }\mathrm{1}}}}}$$\mathit{\mathrm{Re}\left ( s \right )> -a}$
$\mathrm{\mathit{\mathrm{sin}\: \omega t\: u \left ( t \right )}}$$\mathrm{\mathit {\frac{\omega }{\left ( s^{\mathrm{2}}\mathrm{+ }\omega ^{\mathrm{2}} \right )}}}$$\mathit{\mathrm{Re}\left ( s \right )> \mathrm{0}}$
$\mathrm{\mathit{\mathrm{cos}\: \omega t\: u \left ( t \right )}}$$\mathrm{\mathit {\frac{s }{\left ( s^{\mathrm{2}}\mathrm{+ }\omega ^{\mathrm{2}} \right )}}}$$\mathit{\mathrm{Re}\left ( s \right )> \mathrm{0}}$
$\mathrm{\mathit{e^{-at}\: \mathrm{sin}\: \omega t\: u \left ( t \right )}}$$\mathrm{\mathit {\frac{\omega }{\left ( s\mathrm{+ }a \right )^{\mathrm{2}}\mathrm{+ }\omega ^{\mathrm{2}}}}}$$\mathit{\mathrm{Re}\left ( s \right )> -a}$
$\mathrm{\mathit{e^{-at}\: \mathrm{cos}\: \omega t\: u \left ( t \right )}}$$\mathrm{\mathit {\frac{\left ( s\mathrm{+ }a \right )}{\left ( s\mathrm{+ }a \right )^{\mathrm{2}}\mathrm{+ }\omega ^{\mathrm{2}}}}}$$\mathit{\mathrm{Re}\left ( s \right )> -a}$
$\mathrm{\mathit{\mathrm{sin}\left ( \omega t\mathrm{+ }\theta \right )}}$$\mathrm{\mathit {\frac{s\: \mathrm{sin}\, \theta \mathrm{+ }\omega \: \mathrm{cos}\, \theta }{\left ( s^{\mathrm{2}}\mathrm{+ }\omega^{\mathrm{2}} \right )}}}$$$\mathit{\mathrm{Re}\left ( s \right )> \mathrm{0}}\mathrm{\mathit{\mathrm{cos}\left ( \omega t\mathrm{+ }\theta \right )}}$$\mathrm{\mathit {\frac{s\: \mathrm{cos}\, \theta \mathrm{+ }\omega \: \mathrm{sin}\, \theta }{\left ( s^{\mathrm{2}}\mathrm{+ }\omega^{\mathrm{2}} \right )}}}$$\mathit{\mathrm{Re}\left ( s \right )> \mathrm{0}}\mathrm{\mathit t\: {\mathrm{sin}\:\mathit{\omega t\: u\left ( t \right )}}}$$\mathrm{\mathit {\frac{\mathrm{2}\omega s}{\left ( s^{\mathrm{2}}\mathrm{+ }\omega^{\mathrm{2}} \right )\mathrm{^{2}}}}}$$\mathit{\mathrm{Re}\left ( s \right )> \mathrm{0}}\mathrm{\mathit t\: {\mathrm{cos}\:\mathit{\omega t\: u\left ( t \right )}}}$$\mathrm{\mathit {\frac{ s^{\mathrm{2}}-\omega ^{\mathrm{2}}}{\left ( s^{\mathrm{2}}\mathrm{+ }\omega^{\mathrm{2}} \right )\mathrm{^{2}}}}}$$\mathit{\mathrm{Re}\left ( s \right )> \mathrm{0}}\mathrm{\mathit {\mathrm{sinh}\:\mathit{\omega t\: u\left ( t \right )}}}$$\mathrm{\mathit {\frac{ \omega }{\left ( s^{\mathrm{2}}-\omega^{\mathrm{2}} \right )}}}$$\mathit{\mathrm{Re}\left ( s \right )> \omega }\mathrm{\mathit {\mathrm{cosh}\:\mathit{\omega t\: u\left ( t \right )}}}$$\mathrm{\mathit {\frac{ s }{\left ( s^{\mathrm{2}}-\omega^{\mathrm{2}} \right )}}}$$\mathit{\mathrm{Re}\left ( s \right )> \omega }\mathrm{\mathit {e^{-at}\: \mathrm{sinh}\:\mathit{\omega t\: u\left ( t \right )}}}$$\mathrm{\mathit {\frac{ \omega }{\left ( s\mathrm{+ }a\right )^{\mathrm{2}}-\omega^{\mathrm{2}}}}}$$\mathit{\mathrm{Re}\left ( s \right )> \left ( \omega -a \right ) }\mathrm{\mathit {e^{-at}\: \mathrm{cosh}\:\mathit{\omega t\: u\left ( t \right )}}}$$\mathrm{\mathit {\frac{ s\mathrm{+ }a }{\left ( s\mathrm{+ }a\right )^{\mathrm{2}}-\omega^{\mathrm{2}}}}}$$\mathit{\mathrm{Re}\left ( s \right )> \left ( \omega -a \right ) }\$