Extension of Fourier Transform - Laplace Transform

The Laplace transform has many applications in physics and engineering. The way it works is to use a linear operator of a function f(t) with a real argument t (t ≥ 0) that transforms f(t) to a function F(s) with complex argument s.We use this transformation for the majority of practical uses; the most-common pairs of f(t) and F(s) are often given in tables for easy reference. The theory was named after Pierre-Simon Laplace (/ləˈplɑːs/), who introduced the transform in his work on probability theory.The Laplace transform is related to the Fourier transform, but the differences is that the Fourier transform expresses a function or signal as a superposition of sinusoids, the Laplace transform expresses in term of function. The similarity between Fourier transform and the Laplace transform is that both are used for solving differential and integral equations. In physics and engineering it is used for analysis of linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems. Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system. Fourier analysis is more focused on steady-state response and for Laplace transform we focus on causal signals x(t) = 0 for t<0, so we use the one-sided LT.

Given a signal, the set of all complex member s for which the integral exists is the region of convergence.

Youtube video: Laplace transform ( From video 26 all the way to 39). Laplace transform graphical

Properties of Laplace transform
Youtube: properties of Laplace transform

1. Linearity

We use the general form of linearity which is show below And the prove for the linearity in Laplace transform is listed below

2. Time delay

Time delay need some subtleties involved in understanding how to apply it. Which is show below: Always remember to go back the original definition of Laplace transform to prove a property. Later we can change the lower limit to different value and do a time shift for the integral. Always keep in mind that to apply the time delay property you must multiply a delayed version of your function by a delayed step.

3. Derivative (1st derivative, 2nd derivative, and Nth derivative)

First derivative:

Below show the derivative property of the Laplace transform And the prove for this property is listed below second derivative: (not that much different from first derivative) For derivative in general follow the rule listed below 4. Integration

Below show the integration property and the prove of the property Final form for integration property is listed below : 5. Convolution

Look back to old lectures if you forget about what convolution is. Before we do convolution we need to assume all the function are causal And below is the when the definition of Laplace transform apply to the property And below is the prove of the convolution property

6. Initial value theorem

First is the initial equation for the property And it start with the differentiation property And the proof for the differentiation with the initial value theorem And final product is 7. Final value theorem

As usual we list the property: And the proof for this property is listed below Final product

Most of the Fourier Transform property still hold in Laplace transform

8. Multiplication 9. Complex shift 10. Time scaling Inverse Laplace transform

Practice Problems
Some example of Laplace transform

MIT Laplace transform question: Question Solution

More: question solution

Another one: Question and solution included

Practice question on pure laplace transform:Question

Extra Resources
Basic definition of Laplace transform : http://www.sosmath.com/diffeq/laplace/basic/basic.html

MIT course: Lecture and video

Very useful website: I use this to survive all my math classes http://tutorial.math.lamar.edu/Classes/DE/LaplaceIntro.aspx

Table for Laplace transform http://tutorial.math.lamar.edu/Classes/DE/Laplace_Table.aspx