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 two-sided Laplace transform or bilateral Laplace transform is an
integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in ...
equivalent to
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
's
moment generating function
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compare ...
. Two-sided Laplace transforms are closely related to the
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
, the
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 ...
, the
Z-transform
In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation.
It can be considered as a discrete-tim ...
and the ordinary or one-sided
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 ...
. If ''f''(''t'') is a real- or complex-valued function of the real variable ''t'' defined for all real numbers, then the two-sided Laplace transform is defined by the integral
:
The integral is most commonly understood as an
improper integral
In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number or positive or negative infinity; or in some instances as both endpoin ...
, which converges if and only if both integrals
:
exist. There seems to be no generally accepted notation for the two-sided transform; the
used here recalls "bilateral". The two-sided transform
used by some authors is
:
In pure mathematics the argument ''t'' can be any variable, and Laplace transforms are used to study how
differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
s transform the function.
In
science
Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earliest archeological evidence for ...
and
engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
applications, the argument ''t'' often represents time (in seconds), and the function ''f''(''t'') often represents a
signal
In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The ''IEEE Transactions on Signal Processing'' ...
or waveform that varies with time. In these cases, the signals are transformed by
filters
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component tha ...
, that work like a mathematical operator, but with a restriction. They have to be causal, which means that the output in a given time ''t'' cannot depend on an output which is a higher value of ''t''.
In population ecology, the argument ''t'' often represents spatial displacement in a dispersal kernel.
When working with functions of time, ''f''(''t'') is called the time domain representation of the signal, while ''F''(''s'') is called the s-domain (or ''Laplace domain'') representation. The inverse transformation then represents a ''synthesis'' of the signal as the sum of its frequency components taken over all frequencies, whereas the forward transformation represents the ''analysis'' of the signal into its frequency components.
Relationship to the Fourier transform
The
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
can be defined in terms of the two-sided Laplace transform:
:
Note that definitions of the Fourier transform differ, and in particular
:
is often used instead. In terms of the Fourier transform, we may also obtain the two-sided Laplace transform, as
:
The Fourier transform is normally defined so that it exists for real values; the above definition defines the image in a strip
which may not include the real axis where the Fourier transform is supposed to converge.
This is then why Laplace transforms retain their value in control theory and signal processing: the convergence of a Fourier transform integral within its domain only means that a linear, shift-invariant system described by it is stable or critical. The Laplace one on the other hand will somewhere converge for every impulse response which is at most exponentially growing, because it involves an extra term which can be taken as an exponential regulator. Since there are no superexponentially growing linear feedback networks, Laplace transform based analysis and solution of linear, shift-invariant systems, takes its most general form in the context of Laplace, not Fourier, transforms.
At the same time, nowadays Laplace transform theory falls within the ambit of more general
integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in ...
s, or even general
harmonical analysis. In that framework and nomenclature, Laplace transforms are simply another form of Fourier analysis, even if more general in hindsight.
Relationship to other integral transforms
If ''u'' is the
Heaviside step function
The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
, equal to zero when its argument is less than zero, to one-half when its argument equals zero, and to one when its argument is greater than zero, then the Laplace transform
may be defined in terms of the two-sided Laplace transform by
:
On the other hand, we also have
:
where
is the function that multiplies by minus one (
), so either version of the Laplace transform can be defined in terms of the other.
The
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 ...
may be defined in terms of the two-sided Laplace transform by
:
with
as above, and conversely we can get the two-sided transform from the Mellin transform by
:
The
moment-generating function
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compare ...
of a continuous
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
''ƒ''(''x'') can be expressed as
.
Properties
The following properties can be found in and
Most properties of the bilateral Laplace transform are very similar to properties of the unilateral Laplace transform,
but there are some important differences:
Parseval's theorem and Plancherel's theorem
Let
and
be functions with bilateral Laplace transforms
and
in the strips of convergence
.
Let
with