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Relationship Of Laplace Transform To Other Transform

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					  Relationship Of Laplace Transform To Other Transform

Relationship Of Laplace Transform To Other Transform

The Laplace transform is a widely used integral transform with many applications in
physics and engineering. Denoted , it is a linear operator of a function f(t) with a real
argument t (t ≥ 0) that transforms it to a function F(s) with a complex argument s. This
transformation is essentially bijective for the majority of practical uses; the respective
pairs of f(t) and F(s) are matched in tables.

The Laplace transform has the useful property that many relationships and operations
over the originals f(t) correspond to simpler relationships and operations over the
images F(s). It is named for Pierre-Simon Laplace, who introduced the transform in
his work on probability theory.

The Laplace transform is related to the Fourier transform, but whereas the Fourier
transform expresses a function or signal as a series of modes of vibration (frequencies),
the Laplace transform resolves a function into its moments.

Like the Fourier transform, the Laplace transform is used for solving differential and
integral equations.
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In physics and engineering it is used for analysis of linear time-invariant systems such as
electrical circuits, harmonic oscillators, optical devices, and mechanical systems.

In such analyses, the Laplace transform is often interpreted as a transformation from the
time-domain, in which inputs and outputs are functions of time, to the frequency-domain,
where the same inputs and outputs are functions of complex angular frequency, in
radians per unit time.

Given a simple mathematical or functional description of an input or output to a system,
the Laplace transform provides an alternative functional description that often simplifies
the process of analyzing the behavior of the system, or in synthesizing a new system
based on a set of specifications.

Laplace transform rechecks a function into its instance. Relationship of Laplace
Transform to other transform can be given as:

Laplace transform performs on the density of function of measure while Laplace stieltjes
transform do work on its cumulative distribution function.

The relationship between Fourier and Laplace transform can be used to find dynamical
system or frequency spectrum of a signal.

Mellin transform is related to two sided Laplace transform or bilateral transform by a
simple change of variables and z-transform is related to one sided or unilateral transform
of an ideally sampled signal. Borel transform is used for entire function of exponential
type.

Fourier Transform :- Fourier transform is a function of mathematics that is used in
various fields like engineering and physics. Fourier transform is a function of time that is
related to the frequency known as frequency spectrum.
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F^ is the common convention that is an integral function f: R->C.
Fourier transform is basically described by the study of the Fourier series that.

Laplace–Stieltjes transform

Laplace Stieltjes transform is a form of integral transform like Laplace Transform named
after the scientists Pierre-Simon Laplace and Thomas Joannes Stieltjes. It is often
defined for the functions with values in Banach space. It’s an important concept of
mathematics and used in a number of areas like function analysis, applied probability etc.

Mellin Transform

Mellin transform is an integral transform in mathematics that can be known as the
multiplicative version of the two sides Laplace Transform. This is used in number theory
and closely related to Fourier and Laplace transform. Mellin transform is named after the
name of great mathematician, ‘Hjalmar Mellin’. The Mellin transform is very useful in
computer science because of its scale invariance property i.e. useful in image recognition.

Z-Transform

Z-transform is similar to the Laplace Transform. Z-transform is the most necessary tool
that helps in system design and analysis and it also inspect the system's stability.

The functionality of z transform provides access insight into the changeable behavior or
transient behavior monitors the stability of discrete time system and the steady state
behavior.




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