Let f of x be defined on the interval. I That means X is between 0 and Infinity. Then we Define Laplace transform Of our function, as the integral between 0 and Infinity of f of x times e to minus X DX. We assume that F is a function for which the integral on the right exists. For some values of S, We can give examples of the Laplace transform, for instance, for f of x equals 1. The Laplace transform is 1 over s. For if f of x is X, the Laplace transform in this case is 1 over s Square. We can prove some properties of the Laplace transform If we assume that Laplace transform Of some function, f of one Converges, for some s bigger than S1 And another function, F2 of X converges for s bigger than S2. Then we can talk about Laplace transform of their linear combination and its converges for s bigger than S1 and S2 This is the linearity. We can generalize this linearity to more than one function, we will need it. When we have to deal with Differential equations, because our aim is to khanacademy telbot cybermath michael penn blackpenredpen solve,khanacademy telbot cybermath michael penn blackpenredpen differential equations, taking care of the initial conditions. And Laplace transform is a quick way to solve them. We can also talk about the inverse Laplace transform. And in that case, we said that Laplace transform is a linear operator.

We will need fractional decomposition And the knowledge of improper integral to be able to solve this differential equations because we have to find the inverse and finding the errors. Requires a knowledge of partial fractional decomposition and improper integral in general. We will we will use Faltung theorem .It's f-a-l-t-u-n-g theorem that deals with Laplace transform.of 2 finctions.. Anyways, And this means that we can define the convolution theorem for this case. And we're gonna see that one. We have two functions. We can use this theorem to find more And later, we're gonna see that we can have a table for Laplace transforms and from there, we can get all the functions that we need later. We're gonna talk about the gamma function. The gamma function is very useful when dealing with transforms, It's very useful because it helps us find the values that we need.
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