Definition:Laplace Transform

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Definition

Let $f: \R_{\ge 0} \to \mathbb F$ be a function of a real variable $t$, where $\mathbb F \in \set {\R, \C}$.

The Laplace transform of $f$, denoted $\laptrans f$ or $F$, is defined as:

$\laptrans {\map f t} = \map F s = \displaystyle \int_0^{\to +\infty} e^{-s t} \map f t \rd t$

whenever this improper integral converges.

If this improper integral does not converge, then $\laptrans {\map f t}$ does not exist.


Here $\laptrans f$ is a complex function of the variable $s$.


Discontinuity at Zero

Let $f: \R_{> 0} \to \mathbb F$ be a function of a real variable $t$, where $\mathbb F \in \set {\R, \C}$.

Let $f$ be discontinuous or not defined at $t = 0$.

Then the Laplace transform of $f$ is defined as:

$\displaystyle \laptrans {\map f t} = \map F s = \int_{0^+}^{\to +\infty} e^{-s t} \map f t \rd t = \lim_{\epsilon \mathop \to 0^+} \int_\epsilon^{\to +\infty} e^{-s t} \map f t \rd t$

whenever this improper integral converges.

If this improper integral does not converge, then $\laptrans {\map f t}$ does not exist.


Notation

The function which serves as the argument of a Laplace transform is usually denoted by means of a lowercase letter, for example $f$, $g$, $y$, and so on.

The Laplace transform of this function is then denoted by the corresponding uppercase letter, that is $F$, $G$, $Y$, and so on.

Hence we have:

$\laptrans {\map f t} = \map F s$


However, note that some sources reverse the cases of the symbols used to denote the functions under discussion:

$\laptrans {\map F t} = \map f s$


Notation for the Laplace transform varies throughout the literature.


The notation preferred on $\mathsf{Pr} \infty \mathsf{fWiki}$ is:

$\laptrans {\map f t} = \map F s$


Other notation that can be seen includes:

  • $\mathcal L \sqbrk {\map f t}$
  • $\mathscr L \set {\map f t}$
  • $\mathbf L \map f t$


It is sometimes worth stressing the point that $\laptrans {\map f t}$ is a function of $s$ by expressing it as:

$\map {\laptrans {\map f t} } s$

and this notation is occasionally seen on $\mathsf{Pr} \infty \mathsf{fWiki}$.


Some sources use a tilde $\tilde f$ to denote the Laplace transform.

Thus the Laplace transform of $\map u t$ is denoted $\map {\tilde u} t$.

However, this usage is discouraged on $\mathsf{Pr} \infty \mathsf{fWiki}$ because the tilde does not present well in the version of the $\LaTeX$ renderer used on $\mathsf{Pr} \infty \mathsf{fWiki}$.


Graphical Interpretation

Define $\gamma$ as the integrand of $\displaystyle \int_0^{\to +\infty} e^{-s t} \map f t \rd t$ as a function of $\map f t$, $s$, and $t$:

$\map \gamma {\map f t, t; s} = \map f t \, e^{-s t}$

For any particular function $f$, holding $s$ fixed, the integrand of the Laplace Transform $\kappa$ can be interpreted as a contour.

That is, for a given function $f$ and a particular complex number $s_0$ held constant:

$\map \kappa t: \R_{\ge 0} \to \C$
$\map \kappa t = \map \gamma {\map f t, t; s_0}$

is a parameterization of a contour.


Restriction to Reals

Although the definition of the Laplace transform has $s$ be a complex variable, sometimes the restriction of $\map {\laptrans f} s$ to wholly real $s$ is sufficient to solve a particular differential equation.

Therefore, elementary textbooks introducing the Laplace transform will often write something like the following:


... where we assume at present that the parameter $s$ is real. Later it will be found useful to consider $s$ complex.
-- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms: Chapter $1$: The Laplace Transform: Definition of the Laplace Transform


A profound understanding of the workings of the Laplace transform requires considering it to be a so-called analytic function of a complex variable, but in most of this book we shall assume that the variable $s$ is real.
-- 2003: Anders Vretblad: Fourier Analysis and its Applications: $\S 3.1$


Examples

Example $1$

$\laptrans {4 e^{5 t} + 6 t^3 - 3 \sin 4 t + 2 \cos 2 t} = \dfrac 4 {s - 5} + \dfrac {36} {s^4} - \dfrac {12} {s^2 + 16} + \dfrac {2 s} {s^2 + 4}$


Also see

  • Results about Laplace transforms can be found here.


Applications in Physics


Source of Name

This entry was named for Pierre-Simon de Laplace.


Historical Note

Despite the fact that the Laplace transform bears the name of Pierre-Simon de Laplace, they were first used by Leonhard Paul Euler to solve differential equations.

Also during the course of solutions of differential equations, Oliver Heaviside advanced sophisticated applications of this transform.


Technical Note

The $\LaTeX$ code for \(\laptrans {\map f t}\) is \laptrans {\map f t} .


Sources