Laplace transform

Laplace transform is a transformational method to convert very complex differential equations into simpler polynomial forms. These polynomial forms contain multiple domain variables. It has a wide array of uses in science and engineering.

The Laplace transform

The Laplace transform can be used to convert a real-variable function into a complex-variable function (seen in the equation below) or differential equations into algebraic ones:

The $s$ seen in this equation is the complex variable, and as with all other complex numbers, it can be defined as follows:

Since we integrate a function over the interval $[0, \infin]$, it is necessary to be able to incorporate different sorts of functions easily, such as, $cosine, sine, differential, geometric$, and so on.

Example

For $e^t$

Let's find the Laplace transform of the following equation:

The equation of conversion would go as follows:

Further evaluating the above would become:

Hence the Laplace transform would be:

For $e^at$

To find the Laplace transform of $e^at$, we can write:

Applications

Some applications of the Laplace theorem are discussed below:

• Calculate output of linear-time invariant systems

• Simplify system behavioral analysis

• Evaluation of improper integrals

• Practical fraction expansion

• Statistical mechanics

• Send signals in telecommunication medium

• Electrical circuit analysis and modeling