# Laplace Series

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```							                              Laplace Series
Laplace Series

In Fourier series and Laplace series first we will discuss about the topic of Fourier series,
and later we will go on laplace series, Fourier series was formulated by a Jean-Baptiste
Fourier. he showed that an imaginary periodic function can be written as a sum of cosine
and sine function.

And in other we can say that Fourier series divides or decompose periodic function or
periodic signal into the sum of sine’s and cosines that are also called complex
exponential. The study related to Fourier series is comes under Fourier analysis. Fourier
introduce this series to solve heat equation in a metal plate,

Before Fourier’s work there was no solution to measure heat equation in general way.
Eigen solution is the solution from which the heat source and Fourier was working on
a supervision of cosine and sine wave give a model for difficult heat source and this
supervision is called Fourier series.

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Fourier series have many applications such as electrical engineering, acoustics,
vibration analysis, signal processing, image processing and in many more.

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. 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.

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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.

Now let’s talk about their comparison how they differ to each other. The Laplace
series(LS) is better for the small times or can say that superior for small times and
Fourier series(FS) is better for large times or superior for large times.

First we take series of Fourier and Laplace than transform that and both are used to
solve differential equation but question is arises that which one is best, it is not define
yet both plays an important roles in their respected condition whenever their
corresponding equation, series is comes.

Both the transform Is used for different-2 purpose, Laplace transform is used when
we deal with initial value and the Fourier transform is useful when we deal with
boundary-value problems.

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Thank You

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