Laplace Calculator

Online Laplace Calculator

Calculate Laplace Transform

Enter a function to calculate its Laplace transform

Examples

Laplace Transform

\mathcal{L}\{e^{t/2}\}

\mathcal{L}\{e^{-2t}\sin^2(t)\}

\mathcal{L}\{8\pi\}

\mathcal{L}\{\sin(at)\}

\mathcal{L}\{t^2\}

Inverse Laplace Transform

\mathcal{L}^{-1}\{\frac{1}{s^2 + 1}\}

\mathcal{L}^{-1}\{\frac{s}{s^2 + 4}\}

\mathcal{L}^{-1}\{\frac{1}{s(s+1)}\}

\mathcal{L}^{-1}\{\frac{1}{s^2}\}

\mathcal{L}^{-1}\{\frac{s}{s^2 + a^2}\}

What is Laplace Calculator?

The Laplace Transform is an integral transform used to solve differential equations. It transforms a function of time, f(t), to a function of complex frequency, F(s). The Laplace Transform is a fundamental tool for bridging the time-domain and the frequency-domain in mathematics, physics, and engineering. A Laplace Calculator is a tool, typically software or an online platform, that automates the process of computing the Laplace Transform or Inverse Laplace Transform of mathematical functions. These calculators are designed to handle a wide range of inputs, including standard mathematical expressions and differential equations, and provide the corresponding results in the s-domain or time-domain.

The mathematical definition of Laplace Transform

The Laplace Transform of a function $f(t)$ , defined for $t \ge 0$ , is given by: $F(s) = \mathcal{L}\{f(t)\} = \int_0^\infty f(t) e^{-st} dt$ where:

𝑡 is the time variable (real and non-negative),
𝑠 is a complex number 𝑠 = 𝜎 + 𝑗𝜔 (with real part 𝜎 and imaginary part 𝜔),
$e^{-st}$ is the exponential kernel that weights 𝑓(𝑡) based on 𝑠.

Why use our Laplace Calculator?

Fast and accurate calculations
Easy to use interface
Supports a wide range of functions
Free to use

What is Inverse Laplace Transform?

The Inverse Laplace Transform is a mathematical operation that converts a function in the s-domain (frequency domain) back into its original form in the t-domain (time domain). It reverses the process of the Laplace Transform.

Resources and Tools to learn Laplace Transform?

Laplace Transform Table

Time Domain $f(t)$	Laplace Domain $F(s)=\mathcal{L}\{f(t)\}$	Region of Convergence
$1$	$\frac{1}{s}$	$Re(s) > 0$
$t$	$\frac{1}{s^2}$	$Re(s) > 0$
$t^n$	$\frac{n!}{s^{n+1}}$	$Re(s) > 0$
$e^{at}$	$\frac{1}{s-a}$	$Re(s) > Re(a)$
$\sin(at)$	$\frac{a}{s^2 + a^2}$	$Re(s) > 0$
$\cos(at)$	$\frac{s}{s^2 + a^2}$	$Re(s) > 0$
$t\sin(at)$	$\frac{2as}{(s^2 + a^2)^2}$	$Re(s) > 0$
$t\cos(at)$	$\frac{s^2 - a^2}{(s^2 + a^2)^2}$	$Re(s) > 0$
$e^{at}\sin(bt)$	$\frac{b}{(s-a)^2 + b^2}$	$Re(s) > Re(a)$
$e^{at}\cos(bt)$	$\frac{s-a}{(s-a)^2 + b^2}$	$Re(s) > Re(a)$

Note: This table shows some of the most commonly used Laplace transform pairs. The Region of Convergence (ROC) indicates where the transform is valid in the complex s-plane.

Inverse Laplace Transform Table

Laplace Domain $F(s)$	Time Domain $f(t)=\mathcal{L}^{-1}\{F(s)\}$	Conditions
$\frac{1}{s}$	$1$	$t > 0$
$\frac{1}{s^2}$	$t$	$t > 0$
$\frac{1}{s^{n+1}}$	$\frac{t^n}{n!}$	$t > 0$
$\frac{1}{s-a}$	$e^{at}$	$t > 0$
$\frac{a}{s^2 + a^2}$	$\sin(at)$	$t > 0$
$\frac{s}{s^2 + a^2}$	$\cos(at)$	$t > 0$
$\frac{1}{(s-a)^2}$	$te^{at}$	$t > 0$
$\frac{1}{s^2 + a^2}$	$\frac{1}{a}\sin(at)$	$t > 0$
$\frac{s^2}{(s^2 + a^2)^2}$	$\frac{1}{2}t\sin(at)$	$t > 0$
$\frac{s}{(s^2 + a^2)^2}$	$\frac{t}{2a}\cos(at)$	$t > 0$

Note: This table shows common inverse Laplace transform pairs. The condition t > 0 is required for all inverse transforms as they are only defined for positive time.