Laplace Transform Visually Explained, Part 1: Definition, Qualitative Observations, Basic Examples

Do you want to understand the visual meaning of the Laplace Transform? In this video series on the Laplace Transform, I explain graphical interpretations (visual explanations) in addition to symbolic calculations, both for basic examples and for theoretical results, as well as study applications to ordinary differential equations. The Laplace transform is defined and explained as an operator that takes an input function f(t) and gives an output function F(s). I also explain it visually using graphs made using Mathematica. It is analogous to the derivative operator the indefinite integral operator. For a fixed s, the Laplace transform F(s) is the improper integral of f(t)*e^(-s*t) as t goes from 0 to infinity. When f(t) is non-negative, it gives the area under the graph. The Laplace transform is a decreasing function of s. Four basic examples are considered and graphed in Wolfram Mathematica, making qualitative observations about these graphs. Example 1 is the Laplace transform of the constant function f(t) = 1, which is F(s) = L[1] = 1/s when s is positive. Example 2 is the Laplace transform of f(t) = t, which is F(s) = L[t] = 1/s^2 when s is positive. Example 3 is the Laplace transform of f(t) = t^2, which is F(s) = L[t^2] = 2/s^3 when s is positive. And Example 4 is the Laplace transform of a piecewise linear function f(t) = 1 - t/10 for t between 0 and 10 and f(t) = 0 when t is greater than 10.

Laplace Transform Visually Explained, Part 1. (a.k.a. Introduction to the Laplace Transform, Part 1).

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[deepakmirchandani]

thanks sir for such an informative video with graphs, on laplace transform. great

[vancart]

wow thank you so much. i been looking for detailed laplace transform videos on youtube for loooong time. i dont have time to watch it at the moment but definetly i willl watch it. i hope you talking about applications of laplace transform with details. i would love to listen about that. thank you!

[emiliomesa]

This is some amzaing content, thank you!

[masoudy]

Awesome❤, may i ask what software you used to run the code so i can tedt other problems ? Tnx btw.

[weisanpang]

S being complex, when you said S>0, does that mean the img part of S is 0 ?

[linganiveths.g]

happy evening sir, could u pls make a video on solving a 3rd order or any higher order differential equation, related to electrical circuits
(using laplace transforms)

[Bel...QWERTY]

Please could you add a real physical example to earn the connection between time and frequency by laplace