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History Of Laplace Transform

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					             History Of Laplace Transform
History Of Laplace Transform

Today, we will study an important part of mathematics i.e. laplace transform. Laplace
transform gives us a way to represent linear systems in terms of algebra. Integral
Transform is one of the main applications of the Laplace Transform.

Laplace Transform is denoted by Lf(x) here we have a function f(x) with a value x in
the function on which we are applying a linear operator and we should keep a check
on the value x which must be always greater than or equal to zero(x≥0) the value is
then stored in another function F(a) where “a” is having a value with in a value.

Even if f(x) has very complicated values and it may contain some difficult operations
it all converted into the easy one when it comes to F(a).

Fourier Transform which is an another huge field which deals in the we can say
frequencies of the expression but we will talk about this later on, lets be back to
Laplace which help Fourier to solve their functions having iota(Complex Functions)
into its shape or set of points.
                          Know More About A Real Number That Is Not Rational



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The basic formula of laplace is L f(x)

Laplace transform of a function f(x) is defined for all real numbers x≥0

F(a) = Lf(x) = ∫e^-st f(x)dx

in the example the upperlimit and the lower limit of the integrand is ∞ and 0.

in the above example F(a), a is a complex number

a =p + iq where p and q are real numbers. This is an example of unilateral laplace
trnsform or one sided laplace transform

the only condition is that the function F(a) should be integarble at infinity and both.

 According to probability theory the laplace transform works on expectation value. The
laplace transform is given by

(Lf) (a) = E[e^-aX]

This is known as laplace transform of any random variable a. If we replace a by –x
then we get the function which will generate into its shape or set of points

The laplace transform can be of two types :

1. One sided or unilateral

2. Two sided or bilateral
                               Read  More About Anti Derivative Of Trig Functions



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The example shown above was an example of unilateral transform

Example of bilateral transform is   L(a) = lf(x) =∫f(x) * (e^-ax) dx

Let us look at an example of the above formula.

For f(x) = 5

F(a) = ∫f(x) * e^-ax dx This is the formula of Laplace having Upper limit ∞ and lower
limit 0

F(a)= ∫5 * e^-ax Now we put the value of f(x)

  F(a) = -(5/a)*e^-ax After the Integeration we put the value of upper limit and lower
limit

F(a)= [-(5/a) *e^-a∞ ] - [-(5/a)*e^-a0] Now we put the limits and solve it

F(a) = 5/a the final solution




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posted:6/6/2012
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