Laplace transform
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In mathematics, the Laplace transform, named after its discoverer Pierre-Simon
Laplace (/ləˈplɑːs/), is an integral transform that converts a function of
a real variable (usually , in the time domain) to a function of a complex variable
(in the complex frequency domain, also known as s-domain, or s-plane). The
transform has many applications in science and engineering because it is a tool for
solving differential equations.[1] In particular, it transforms ordinary differential
equations into algebraic equations and convolution into multiplication.[2][3] For suitable
functions f, the Laplace transform is the integral
Contents
1History
2Formal definition
o 2.1Bilateral Laplace transform
o 2.2Inverse Laplace transform
o 2.3Probability theory
3Region of convergence
4Properties and theorems
o 4.1Relation to power series
o 4.2Relation to moments
o 4.3Computation of the Laplace transform of a function's derivative
o 4.4Evaluating integrals over the positive real axis
5Relationship to other transforms
o 5.1Laplace–Stieltjes transform
o 5.2Fourier transform
o 5.3Mellin transform
o 5.4Z-transform
o 5.5Borel transform
o 5.6Fundamental relationships
6Table of selected Laplace transforms
7s-domain equivalent circuits and impedances
8Examples and applications
o 8.1Evaluating improper integrals
o 8.2Complex impedance of a capacitor
o 8.3Partial fraction expansion
o 8.4Phase delay
o 8.5Statistical mechanics
o 8.6Spatial (not time) structure from astronomical spectrum
9Gallery
10See also
11Notes
12References
o 12.1Modern
o 12.2Historical
13Further reading
14External links
History[edit]
Pierre-Simon, marquis de Laplace
The Laplace transform is named after mathematician and astronomer Pierre-Simon,
marquis de Laplace, who used a similar transform in his work on probability theory.
[4]
Laplace wrote extensively about the use of generating functions in Essai
philosophique sur les probabilités (1814), and the integral form of the Laplace transform
evolved naturally as a result.[5]
Laplace's use of generating functions was similar to what is now known as the z-
transform, and he gave little attention to the continuous variable case which was
discussed by Niels Henrik Abel.[6] The theory was further developed in the 19th and early
20th centuries by Mathias Lerch,[7] Oliver Heaviside,[8] and Thomas Bromwich.[9]
The current widespread use of the transform (mainly in engineering) came about during
and soon after World War II,[10] replacing the earlier Heaviside operational calculus. The
advantages of the Laplace transform had been emphasized by Gustav Doetsch,[11] to
whom the name Laplace transform is apparently due.
From 1744, Leonhard Euler investigated integrals of the form
as solutions of differential equations, but did not pursue the matter very far. [12] Joseph
Louis Lagrange was an admirer of Euler and, in his work on integrating probability
density functions, investigated expressions of the form
which some modern historians have interpreted within modern Laplace transform
theory.[13][14][clarification needed]
These types of integrals seem first to have attracted Laplace's attention in 1782, where
he was following in the spirit of Euler in using the integrals themselves as solutions of
equations.[15] However, in 1785, Laplace took the critical step forward when, rather than
simply looking for a solution in the form of an integral, he started to apply the transforms
in the sense that was later to become popular. He used an integral of the form
akin to a Mellin transform, to transform the whole of a difference equation, in order to
look for solutions of the transformed equation. He then went on to apply the Laplace
transform in the same way and started to derive some of its properties, beginning to
appreciate its potential power.[16]
Laplace also recognised that Joseph Fourier's method of Fourier series for solving
the diffusion equation could only apply to a limited region of space, because those
solutions were periodic. In 1809, Laplace applied his transform to find solutions that
diffused indefinitely in space.[17]
Formal definition[edit]
The Laplace transform of a function f(t), defined for all real numbers t ≥ 0, is the
function F(s), which is a unilateral transform defined by
(Eq.1)
where s is a complex number frequency parameter
with real numbers σ and ω.
An alternate notation for the Laplace transform is instead of F.[3]
The meaning of the integral depends on types of functions of interest. A necessary
condition for existence of the integral is that f must be locally integrable on [0, ∞). For
locally integrable functions that decay at infinity or are of exponential type ( ), the
integral can be understood to be a (proper) Lebesgue integral. However, for many
applications it is necessary to regard it as a conditionally convergent improper
integral at ∞. Still more generally, the integral can be understood in a weak sense, and
this is dealt with below.
One can define the Laplace transform of a finite Borel measure μ by the Lebesgue
integral[18]
An important special case is where μ is a probability measure, for example, the Dirac
delta function. In operational calculus, the Laplace transform of a measure is often
treated as though the measure came from a probability density function f. In that case,
to avoid potential confusion, one often writes
where the lower limit of 0− is shorthand notation for
This limit emphasizes that any point mass located at 0 is entirely captured by the
Laplace transform. Although with the Lebesgue integral, it is not necessary to take such
a limit, it does appear more naturally in connection with the Laplace–Stieltjes transform.
Bilateral Laplace transform[edit]
Main article: Two-sided Laplace transform
When one says "the Laplace transform" without qualification, the unilateral or one-sided
transform is usually intended. The Laplace transform can be alternatively defined as
the bilateral Laplace transform, or two-sided Laplace transform, by extending the limits
of integration to be the entire real axis. If that is done, the common unilateral transform
simply becomes a special case of the bilateral transform, where the definition of the
function being transformed is multiplied by the Heaviside step function.
The bilateral Laplace transform F(s) is defined as follows:
(Eq.2)
An alternate notation for the bilateral Laplace transform is , instead of .
