Laplace Transform History Laplace Transform History 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. The basic formula of laplace is L f(x)
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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 The example shown above was an example of unilateral transform Example of bilateral transform is
L(a) = lf(x) =∫f(x) * (e^-ax) dx
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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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