# Table of Laplace Transforms

## Related calculators: Laplace Transform Calculator , Inverse Laplace Transform Calculator

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:

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