# Category: Laplace Transform

## Definition of the Laplace Transform

The Laplace transform of a function ${f{{\left({t}\right)}}}$, defined for all ${t}\ge{0}$, is the function ${F}{\left({s}\right)}$, defined as follows:

${F}{\left({s}\right)}={L}{\left({f{{\left({t}\right)}}}\right)}={\int_{{0}}^{\infty}}{{e}}^{{-{s}{t}}}{f{{\left({t}\right)}}}{d}{t}$, where ${s}$ is a complex parameter.

## Table of Laplace Transforms

This is not a complete list of Laplace transforms, but it contains all common transforms, which can be used to quickly find solutions of differential equations and integrals:

${f{{\left({t}\right)}}}={{L}}^{{-{1}}}{\left({F}{\left({s}\right)}\right)}$ ${F}{\left({s}\right)}={L}{\left({f{{\left({t}\right)}}}\right)}$ ${1}$ $\frac{{1}}{{s}}$ ${{t}}^{{n}}$, ${n}={0},{1},{2},{3}\ldots$ $\frac{{{n}!}}{{{{s}}^{{{n}+{1}}}}}$ ${{t}}^{{n}}$, ${n}>-{1}$ $\frac{{\Gamma{\left({n}+{1}\right)}}}{{{s}}^{{{n}+{1}}}}$ ${{e}}^{{{a}{t}}}$ $\frac{{1}}{{{s}-{a}}}$ ${{t}}^{{{n}-\frac{{1}}{{2}}}}$, ${n}={1},{2},{3}\ldots$ $\frac{{{1}\cdot{3}\cdot{5}\cdot\ldots\cdot{\left({2}{n}-{1}\right)}\cdot\sqrt{{\pi}}}}{{{{2}}^{{n}}{{s}}^{{{n}+\frac{{1}}{{2}}}}}}$ $\sqrt{{{t}}}$ $\frac{\sqrt{{\pi}}}{{{2}{{s}}^{{\frac{{3}}{{2}}}}}}$ ${\sin{{\left({a}{t}\right)}}}$ $\frac{{a}}{{{{s}}^{{2}}+{{a}}^{{2}}}}$ ${\cos{{\left({a}{t}\right)}}}$ $\frac{{s}}{{{{s}}^{{2}}+{{a}}^{{2}}}}$ ${\sinh{{\left({a}{t}\right)}}}$ $\frac{{a}}{{{{s}}^{{2}}-{{a}}^{{2}}}}$ ${\cosh{{\left({a}{t}\right)}}}$ $\frac{{s}}{{{{s}}^{{2}}-{{a}}^{{2}}}}$ ${t}{\sin{{\left({a}{t}\right)}}}$ $\frac{{{2}{a}{s}}}{{{\left({{s}}^{{2}}+{{a}}^{{2}}\right)}}^{{2}}}$ ${t}{\cos{{\left({a}{t}\right)}}}$ $\frac{{{{s}}^{{2}}-{{a}}^{{2}}}}{{{\left({{s}}^{{2}}+{{a}}^{{2}}\right)}}^{{2}}}$ ${\sin{{\left({a}{t}+{b}\right)}}}$ $\frac{{{s}\cdot{\sin{{\left({b}\right)}}}+{a}\cdot{\cos{{\left({b}\right)}}}}}{{{{s}}^{{2}}+{{a}}^{{2}}}}$ ${\cos{{\left({a}{t}+{b}\right)}}}$ $\frac{{{s}\cdot{\cos{{\left({b}\right)}}}-{a}\cdot{\sin{{\left({b}\right)}}}}}{{{{s}}^{{2}}+{{a}}^{{2}}}}$ ${{e}}^{{{a}{t}}}{\sin{{\left({b}{t}\right)}}}$ $\frac{{b}}{{{{\left({s}-{a}\right)}}^{{2}}+{{b}}^{{2}}}}$ ${{e}}^{{{a}{t}}}{\cos{{\left({b}{t}\right)}}}$ $\frac{{{s}-{a}}}{{{{\left({s}-{a}\right)}}^{{2}}+{{b}}^{{2}}}}$ ${{e}}^{{{a}{t}}}{\sinh{{\left({b}{t}\right)}}}$ $\frac{{b}}{{{{\left({s}-{a}\right)}}^{{2}}-{{b}}^{{2}}}}$ ${{e}}^{{{a}{t}}}{\cosh{{\left({b}{t}\right)}}}$ $\frac{{{s}-{a}}}{{{{\left({s}-{a}\right)}}^{{2}}-{{b}}^{{2}}}}$ ${{t}}^{{n}}{{e}}^{{{a}{t}}}$, ${n}={1},{2},{3}\ldots$ $\frac{{{n}!}}{{{\left({s}-{a}\right)}}^{{{n}+{1}}}}$ ${f{{\left({c}{t}\right)}}}$ $\frac{{1}}{{c}}{F}{\left(\frac{{s}}{{c}}\right)}$ ${u}_{{c}}{\left({t}\right)}={u}{\left({t}-{c}\right)}$ $\frac{{{e}}^{{-{c}{s}}}}{{s}}$ ${u}_{{c}}{\left({t}\right)}{f{{\left({t}-{c}\right)}}}$ ${{e}}^{{-{c}{s}}}{F}{\left({s}\right)}$ $\delta{\left({t}-{c}\right)}$ ${{e}}^{{-{c}{s}}}$ ${{e}}^{{{c}{t}}}{f{{\left({t}\right)}}}$ ${F}{\left({s}-{c}\right)}$ ${{t}}^{{n}}{f{{\left({t}\right)}}}$, ${n}={1},{2},{3}\ldots$ ${{\left(-{1}\right)}}^{{n}}{{F}}^{{{\left({n}\right)}}}{\left({s}\right)}$ ${\int_{{0}}^{{t}}}{f{{\left(\tau\right)}}}{d}\tau$ $\frac{{{F}{\left({s}\right)}}}{{s}}$ ${\int_{{0}}^{{t}}}{f{{\left({t}-\tau\right)}}}{g{{\left(\tau\right)}}}{d}\tau$ ${F}{\left({s}\right)}{G}{\left({s}\right)}$ ${f{'}}{\left({t}\right)}$ ${s}{F}{\left({s}\right)}-{f{{\left({0}\right)}}}$ ${f{''}}{\left({t}\right)}$ ${{s}}^{{2}}{F}{\left({s}\right)}-{s}{f{{\left({0}\right)}}}-{f{'}}{\left({0}\right)}$ ${{f}}^{{{\left({n}\right)}}}{\left({t}\right)}$ ${{s}}^{{n}}{F}{\left({s}\right)}-{\sum_{{{k}={0}}}^{{{n}-{1}}}}{\left({{s}}^{{{n}-{1}-{k}}}{{f}}^{{{\left({k}\right)}}}{\left({0}\right)}\right)}$

## Properties of the Laplace Transform

The Laplace transform has a number of interesting properties.

Property 1. Linearity of the Laplace transform: ${L}{\left({a}{f{{\left({t}\right)}}}+{b}{g{{\left({t}\right)}}}\right)}={a}{L}{\left({f{{\left({t}\right)}}}\right)}+{L}{\left({g{{\left({t}\right)}}}\right)}$, where ${a}$ and ${b}$ are some constants.

## Inverse Laplace Transform

As has been seen previously, the formula for the Laplace transform is ${F}{\left({s}\right)}={L}{\left({f{{\left({t}\right)}}}\right)}$; in other words, we are given a function ${f{{\left({t}\right)}}}$ and we need to find ${F}{\left({s}\right)}$. The inverse Laplace transform is the operation of finding ${f{{\left({t}\right)}}}$ given ${F}{\left({s}\right)}$.

## Convolution Integral

To find the inverse Laplace transform, partial fraction decomposition is very useful, but sometimes it can be very difficult to find the partial fraction decomposition, so there are cases where the inverse Laplace transform can be found with the help of the convolution integral.

## Unit (Heaviside) Step Function

The Heaviside step function is defined as follows:

${u}_{{c}}{\left({t}\right)}={u}{\left({t}-{c}\right)}={H}{\left({t}-{c}\right)}={\left\{\begin{array}{c}{1}{\quad\text{if}\quad}{t}\ge{c}\\{0}{\quad\text{if}\quad}{t}<{c}\\ \end{array}\right.}$

## Dirac Delta Function

The Heaviside function represents switches from one value to another at some point, but what if we need an instant change to a very big value? This is what the Dirac delta function means.

The Dirac delta function is defined as follows: $\delta{\left({t}-{c}\right)}={\left\{\begin{array}{c}+\infty{\quad\text{if}\quad}{t}={c}\\{0}{\quad\text{if}\quad}{t}\ne{c}\\ \end{array}\right.}$,

## Solving IVPs with Laplace Transform

You've probably asked yourself why the Laplace transform is in the Differential Equations section. The answer is simple: because we can solve initial-value problems with the help of the Laplace transform.