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Prerita Agarwal

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23 Jul, 2024 @ 01:30 PM

Introduction

Laplace Transform is named in honor of Pierre Simon De Laplace(French Mathematician). As the name contains transform, the Laplace Transform changes one signal into another according to the fixed set of rules and regulations.

It has a major role in system engineering. Laplace transforms of different functions have to be carried out to analyze the control system. Both the properties of the Laplace transform, as well as the inverse Laplace transformation are used in analyzing the dynamic control system.

What is the definition of Laplace Transform?

The Laplace transform of a function is represented by L{f(t)} or F(s). It helps to solve the differential equations, where it reduces the differential equation into an algebraic problem.

Laplace Transform Formula:

Properties Of Laplace Transform

We will discuss the derivation of some of the properties of Laplace Transformation. These properties will allow us to solve differential equations and higher-level analyses of algorithms.

Linearity Property

The linearity property of Laplace Transform states that:

We will prove this property with the help of the definition of Laplace Transform:

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Time Delay Property

The time shift property states that:

Again we will prove this property with the help of the definition of Laplace Transform:

Here we can change the lower limit of the integral from 0- to a-and drop the step function (as it will always be equal to one)

The last integral shown is just the definition of the Laplace Transform, so we have the time delay property as:

Note:

To apply the time delay theorem, you should multiply a delayed version of your function by a delayed step. If the originally given function is g(t)Â·Î³(t), then the shifted function is g(t-td)Â·Î³(t-td) -where td is the time delay.

First Derivative

The first derivative property of the Laplace Transform states:

We will prove this property by integration of parts:

The first term in the parenthesis goes to zero. In the next term of the parenthesis, the exponential goes to one. And the last term is simply the definition of the Laplace Transform multiplied by s. Hence the theorem is proved.

Integration Property

The integration property states that:

We will prove this property by integration of parts:

The first term in the bracket goes to zero if f(t) grows more slowly than an exponential, and the second term goes to zero because the limits on the integral are equal. So the theorem is proven.

Convolution Property

The convolution property states that :

We will prove this property with the help of the definition of Laplace Transform:

Initial Value Theorem

The initial value theorem states that:

To show this, we first start with the Derivative Rule:

Several simplifications are in order. In the left-hand expression, we can take the second term out of the limit since it doesn't depend on 's.' In the right-hand expression, we can take the first term out of the limit for the same reason mentioned above, and if we substitute infinity for 's' in the second term, the exponential term will go to zero:

The given two f(0-) terms cancel each other, leaving us with the Initial Value Theorem.

The Laplace transform of a function is represented by L{f(t)} or F(s). It helps to solve the differential equations, where it reduces the differential equation into an algebraic problem.

2. State some applications of Laplace transform.

It is used to convert complex differential equations to a simpler form having polynomials.

It is also used for many tasks such as Digital Signal Processing, System Modelling, Electrical Circuit Analysis, etc.

Key Takeaways

This blog has extensively discussed the Laplace Transform and its properties, along with the derivation of some of the properties. It has a major role in converting the complex differential equations to a simpler form having polynomials.

These functions will help with a huge amount of data, and we want some detailed information.