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

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