# Table of Laplace Transforms

## Definition of Laplace Transforms

 Let $$f(t)$$ be a function of the real variable $$t$$, such that $$t \geq 0$$. The Laplace transform $$F(s)$$ of $$f$$ is given by the integral $F(s) = L(f(t) = \int_{0}^{\infty} e^{-st} f(t) \, dt$ where $$s$$ is a complex variable. $$f(t)$$ is called the original and $$F(s)$$ is called the image function.

## Table of Laplace Transforms

 $$f(t)$$ $$F(s)$$ 1 $$\dfrac{1}{s}$$ $$t$$ $$\dfrac{1}{s^2}$$ $$t^n$$ $$\dfrac{n!}{s^{n+1}}$$, $$(n = 1,2,3...)$$ $$t^{1/2}$$ $$\dfrac{\sqrt{\pi}}{2s^{3/2}}$$ $$t^{-1/2}$$ $$\sqrt{\dfrac{\pi}{s}}$$ $$e^{-at}$$ $$\dfrac{1}{s + a}$$ $$t e^{-at}$$ $$\dfrac{1}{(s + a)^2}$$ $$\sin(at)$$ $$\dfrac{a}{s^2 + a^2}$$ $$t \sin(at)$$ $$\dfrac{2as}{(s^2 + a^2)^2}$$ $$e^{-at} \sin(bt)$$ $$\dfrac{b}{(s + a)^2 + b^2}$$ $$\cos(at)$$ $$\dfrac{s}{s^2 + a^2}$$ $$t \cos(at)$$ $$\dfrac{s^2 - a^2}{(s^2 + a^2)^2}$$ $$e^{-at} \cos(bt)$$ $$\dfrac{s + a}{(s + a)^2 + b^2}$$ $$\sinh(at)$$ $$\dfrac{a}{s^2 - a^2}$$ $$\cosh(at)$$ $$\dfrac{s}{s^2 - a^2}$$ $$1 - \cos(at)$$ $$\dfrac{a^2}{s(s^2 + a^2)}$$ $$\left(\dfrac{2}{t}\right)(t - \cos(at))$$ $$\ln \left( \dfrac{s^2 + a^2}{s^2} \right)$$ $$\left(\dfrac{2}{t}\right)(t - \cosh(at))$$ $$\ln \left( \dfrac{s^2 - a^2}{s^2} \right)$$ $$\left(\dfrac{1}{t}\right)(\sin(at))$$ $$\arctan \left( \dfrac{a}{s} \right)$$

## More references on integrals in calculus

Laplace Transform with Examples and Solutions
Find Area Under Curve
table of integrals
properties of integrals