List of Laplace transforms

Template:Short description The following is a list of Laplace transforms for many common functions of a single variable. The Laplace transform is an integral transform that takes a function of a positive real variable Template:Math (often time) to a function of a complex variable Template:Mvar (complex angular frequency).

Properties

The Laplace transform of a function $f(t)$ can be obtained using the formal definition of the Laplace transform. However, some properties of the Laplace transform can be used to obtain the Laplace transform of some functions more easily.

Linearity

For functions $f$ and $g$ and for scalar $a$, the Laplace transform satisfies

$ℒ\{af(t)+bg(t)\}=aℒ\{f(t)\}+bℒ\{g(t)\}$

and is, therefore, regarded as a linear operator.

Time shifting

The Laplace transform of $f(t-a)u(t-a)$ is $e^{-as}F(s)$.

Frequency shifting

$F(s-a)$ is the Laplace transform of $e^{at}f(t)$.

Explanatory notes

The unilateral Laplace transform takes as input a function whose time domain is the non-negative reals, which is why all of the time domain functions in the table below are multiples of the Heaviside step function, Template:Math.

The entries of the table that involve a time delay Template:Math are required to be causal (meaning that Template:Math). A causal system is a system where the impulse response Template:Math is zero for all time Template:Mvar prior to Template:Math. In general, the region of convergence for causal systems is not the same as that of anticausal systems.

The following functions and variables are used in the table below:

• Template:Math represents the Dirac delta function.
• Template:Math represents the Heaviside step function. Literature may refer to this by other notation, including $1(t)$ or $H(t)$.
• Template:Math represents the Gamma function.
• Template:Math is the Euler–Mascheroni constant.
• Template:Math is a real number. It typically represents time, although it can represent any independent dimension.
• Template:Math is the complex frequency domain parameter, and Template:Math is its real part.
• Template:Math is an integer.
• Template:Math are real numbers.
• Template:Math is a complex number.

Table

Function	Time domain $f(t)={ℒ}^{-1}\{F(s)\}$	Laplace Template:Math-domain $F(s)=ℒ\{f(t)\}$	Region of convergence	Reference
unit impulse	$\delta (t)$	$1$	all Template:Math	inspection
delayed impulse	$\delta (t-\tau )$	$e^{-\tau s}$	Template:Math	time shift of unit impulse
unit step	$u(t)$	$\frac{1}{s}$	Template:Math	integrate unit impulse
delayed unit step	$u(t-\tau )$	$\frac{1}{s}{e}^{-\tau s}$	Template:Math	time shift of unit step
ramp	$t\cdot u(t)$	$\frac{1}{s^2}$	Template:Math	integrate unit impulse twice
Template:Mathth power (for integer Template:Math)	$t^n\cdot u(t)$	$\frac{n!}{s^{n+1}}$	Template:Math (Template:Math)	Integrate unit step Template:Math times
Template:Mathth power (for complex Template:Math)	$t^q\cdot u(t)$	$\frac{\mathrm{\Gamma }(q+1)}{s^{q+1}}$	Template:Math Template:Math
Template:Mathth root	$\sqrt[n]{t}\cdot u(t)$	$\frac{1}{s^{\frac{1}{n}+1}}\mathrm{\Gamma }(\frac{1}{n}+1)$	Template:Math	Set Template:Math above.
Template:Mathth power with frequency shift	$t^n{e}^{-\alpha t}\cdot u(t)$	$\frac{n!}{(s+\alpha {)}^{n+1}}$	Template:Math	Integrate unit step, apply frequency shift
delayed Template:Mathth power with frequency shift	$(t-\tau {)}^n{e}^{-\alpha (t-\tau )}\cdot u(t-\tau )$	$\frac{n!\cdot e^{-\tau s}}{(s+\alpha {)}^{n+1}}$	Template:Math	Integrate unit step, apply frequency shift, apply time shift
exponential decay	$e^{-\alpha t}u(t)$	$\frac{1}{s+\alpha }$	Template:Math	Frequency shift of unit step
two-sided exponential decay (only for bilateral transform)	$e^{-\alpha |t|}$	$\frac{2\alpha }{{\alpha }^2-s^2}$	Template:Math	Frequency shift of unit step
exponential approach	$(1-e^{-\alpha t})\cdot u(t)$	$\frac{\alpha }{s(s+\alpha )}$	Template:Math	Unit step minus exponential decay
sine	$\sin (\omega t)\cdot u(t)$	$\frac{\omega }{s^2+{\omega }^2}$	Template:Math
cosine	$\cos (\omega t)\cdot u(t)$	$\frac{s}{s^2+{\omega }^2}$	Template:Math
hyperbolic sine	$\sinh (\alpha t)\cdot u(t)$	$\frac{\alpha }{s^2-{\alpha }^2}$	Template:Math
hyperbolic cosine	$\cosh (\alpha t)\cdot u(t)$	$\frac{s}{s^2-{\alpha }^2}$	Template:Math
exponentially decaying sine wave	$e^{-\alpha t}\sin (\omega t)\cdot u(t)$	$\frac{\omega }{(s+\alpha {)}^2+{\omega }^2}$	Template:Math
exponentially decaying cosine wave	$e^{-\alpha t}\cos (\omega t)\cdot u(t)$	$\frac{s+\alpha }{(s+\alpha {)}^2+{\omega }^2}$	Template:Math
natural logarithm	$\ln (t)\cdot u(t)$	$\frac{-\ln (s)-\gamma }{s}$	Template:Math
Bessel function of the first kind, of order n	$J_n(\omega t)\cdot u(t)$	$\frac{{(\sqrt{s^2+{\omega }^2}-s)}^n}{{\omega }^n\sqrt{s^2+{\omega }^2}}$	Template:Math (Template:Math)
Error function	$\mathrm{erf}(t)\cdot u(t)$	$\frac{e^{s^2/4}}{s}(1-\mathrm{erf}\left(\frac{s}{2}\right))$	Template:Math

See also

• List of Fourier transforms