The table of inverse Laplace transformations is very useful if you want to compute inverse Laplace of functions. In this post, we will provide here the inverse Laplace table.

Table of Contents

# Inverse Laplace Table

Function f(t) |
Inverse Laplace L^{-1}{f(t)} |

1 | δ(t), Dirac delta function |

1/s | 1 |

1/s^{2} |
t |

1/s^{3} |
t^{2}/2 |

1/s^{4} |
t^{3}/6 |

1/s^{n} |
t^{n-1}/(n-1)!, n=1, 2, 3, … |

$\dfrac{1}{s-a}$ | e^{at} |

$\dfrac{1}{s+a}$ | e^{-at} |

$\dfrac{1}{(s-a)^n}$ | $\dfrac{e^{at} t^{n-1}}{(n-1)!}$ |

$\dfrac{1}{s^2+a^2}$ | $\dfrac{1}{a}$ sin at |

$\dfrac{1}{s^2-a^2}$ | $\dfrac{1}{a}$ sinh at |

$\dfrac{s}{s^2+a^2}$ | cos at |

$\dfrac{s}{s^2-a^2}$ | cosh at |

$\dfrac{1}{(s-a)^2+b^2}$ | $\dfrac{1}{b}$ e^{at} sin bt |

$\dfrac{s-a}{(s-a)^2+b^2}$ | e^{at} cos bt |

$\dfrac{s}{(s^2+a^2)^2}$ | $\dfrac{1}{2a}$ t sin at |

$\dfrac{1}{(s^2+a^2)^2}$ | $\dfrac{1}{2a^3}$ (sin at – at cos at) |

# How to Use Inverse Laplace Table

**Example 1: **Find the inverse Laplace of $\dfrac{s^2-2s+3}{s^3}$.

**Solution:**

At first, we will break the given function into parts as follows:

$\dfrac{s^2-2s+3}{s^3}$ = $\dfrac{s^2}{s^3}- \dfrac{2s}{s^3}+\dfrac{3}{s^3}$ = $\dfrac{1}{s}- 2\dfrac{1}{s^2}+\dfrac{3}{s^3}$.

So the inverse Laplace of the given function will be

= L^{-1}(1/s) – 2 L^{-1}(1/s^{2}) + 3 L^{-1}(1/s^{3}).

Now, using the above table of inverse Laplace, the above

= 1 – 2t + 3 t^{2}/2

= (2 – 4t + 3 t^{2})/2.

**Read More Inverse Laplace**

Inverse Laplace of a constant | Inverse Laplace of 1

How to find inverse Laplace of 1/s

How to find inverse Laplace of 1/s^{2}

This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.