Laplace Transform Equation Formula Latex Code and Summary

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Laplace Transform

Tags: #math #laplace transform

Equation

$$\bar{y}(s)=\mathcal{L}(y(t))=\int^{\infty}_{0} e^{-st} y(t) \mathrm{d}t$$

Latex Code

                                 \bar{y}(s)=\mathcal{L}(y(t))=\int^{\infty}_{0} e^{-st} y(t) \mathrm{d}t

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Introduction

$\bar{y}(s)=\mathcal{L}(y(t))=\int^{\infty}_{0} e^{-st} y(t) \mathrm{d}t$

If original function y(t) is a function defined for t \ge 0, the Laplace transform y(s) is defined by the equation as above.

Original Function y(t)	Laplace Transform y(s)
$\delta (t)$	1
$t^{n}$	$\frac{n!}{s^{n+1}}$
$e^{-at}$	$\frac{1}{(s+a)}$
$\sin \omega t$	$\frac{\omega}{s^{2} + \omega^{2}}$
$\cos \omega t$	$\frac{s}{s^{2} + \omega^{2}}$

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