In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the inverse Laplace transform of a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
''F''(''s'') is the piecewise-
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
and exponentially-restricted
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
function ''f''(''t'') which has the property:
:
where
denotes the
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
.
It can be proven that, if a function ''F''(''s'') has the inverse Laplace transform ''f''(''t''), then ''f''(''t'') is uniquely determined (considering functions which differ from each other only on a point set having
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
zero as the same). This result was first proven by
Mathias Lerch
Mathias Lerch (''Matyáš Lerch'', ) (20 February 1860, Milínov – 3 August 1922, Sušice) was a Czech mathematician who published about 250 papers, largely on mathematical analysis and number theory. He studied in Prague and Berlin, and held t ...
in 1903 and is known as Lerch's theorem.
The
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
and the inverse Laplace transform together have a number of properties that make them useful for analysing
linear dynamical system Linear dynamical systems are dynamical systems whose evaluation functions are linear. While dynamical systems, in general, do not have closed-form solutions, linear dynamical systems can be solved exactly, and they have a rich set of mathematical ...
s.
Mellin's inverse formula
An integral formula for the inverse
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
, called the ''Mellin's inverse formula'', the ''
Bromwich integral'', or the ''
Fourier–
Mellin
Mellin is a village and a former municipality in the district Altmarkkreis Salzwedel, in Saxony-Anhalt, Germany. Since 1 January 2009, it is part of the municipality Beetzendorf
Beetzendorf is a municipality in the district Altmarkkreis Salzwe ...
integral'', is given by the
line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, alt ...
:
:
where the integration is done along the vertical line Re(''s'') = ''γ'' in the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
such that ''γ'' is greater than the real part of all
singularities of ''F''(''s'') and ''F''(''s'') is bounded on the line, for example if the contour path is in the
region of convergence
In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series co ...
. If all singularities are in the left half-plane, or ''F''(''s'') is an
entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...
, then ''γ'' can be set to zero and the above inverse integral formula becomes identical to the
inverse Fourier transform In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that if we know all frequency and phase information ...
.
In practice, computing the complex integral can be done by using the
Cauchy residue theorem
In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well ...
.
Post's inversion formula
Post's inversion formula for
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
s, named after
Emil Post
Emil Leon Post (; February 11, 1897 – April 21, 1954) was an American mathematician and logician. He is best known for his work in the field that eventually became known as computability theory.
Life
Post was born in Augustów, Suwałki Govern ...
,
is a simple-looking but usually impractical formula for evaluating an inverse Laplace transform.
The statement of the formula is as follows: Let ''f''(''t'') be a continuous function on the interval [0, ∞) of exponential order, i.e.
:
for some real number ''b''. Then for all ''s'' > ''b'', the Laplace transform for ''f''(''t'') exists and is infinitely differentiable with respect to ''s''. Furthermore, if ''F''(''s'') is the Laplace transform of ''f''(''t''), then the inverse Laplace transform of ''F''(''s'') is given by
:
for ''t'' > 0, where ''F''
(''k'') is the ''k''-th derivative of ''F'' with respect to ''s''.
As can be seen from the formula, the need to evaluate derivatives of arbitrarily high orders renders this formula impractical for most purposes.
With the advent of powerful personal computers, the main efforts to use this formula have come from dealing with approximations or asymptotic analysis of the Inverse Laplace transform, using the Grunwald–Letnikov differintegral to evaluate the derivatives.
Post's inversion has attracted interest due to the improvement in computational science and the fact that it is not necessary to know where the Pole (complex analysis), poles of ''F''(''s'') lie, which make it possible to calculate the asymptotic behaviour for big ''x'' using inverse
Mellin transform In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is
often used i ...
s for several arithmetical functions related to the
Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
.
Software tools
InverseLaplaceTransformperforms symbolic inverse transforms in
Mathematica
Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimizat ...
Numerical Inversion of Laplace Transform with Multiple Precision Using the Complex Domainin Mathematica gives numerical solutions
performs symbolic inverse transforms in
MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation ...
Numerical Inversion of Laplace Transforms in MatlabNumerical Inversion of Laplace Transforms based on concentrated matrix-exponential functionsin Matlab
See also
*
Inverse Fourier transform In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that if we know all frequency and phase information ...
*
Poisson summation formula
In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a ...
References
Further reading
*
*
* (p. 662 or search Index for "Bromwich Integral", a nice explanation showing the connection to the Fourier transform)
*
Elementary inversion of the Laplace transform Bryan, Kurt. Accessed June 14, 2006.
External links
at EqWorld: The World of Mathematical Equations.
{{PlanetMath attribution, id=5877, title=Mellin's inverse formula
Transforms
Complex analysis
Integral transforms
Laplace transforms