What are Inverse Laplace Transforms?
Channel: MechanicaLEi
Date: 2018-07-02

Welcome back MechanicaLEi, Did you know that Thomas John I ‘Anson Bromwich born in 1875 first used Inverse Laplace transforms to justify Operator Calculus? This makes us wonder, what are Inverse Laplace Transforms? Before we jump in check out the previous part of this series to learn about What is Dirac Delta function? Now, If the Laplace transform of a function g of t is the piecewise-continuous and exponentially-restricted real function capital G of s then inverse Laplace transform of capital G of s denoted as L to the power minus one, capital G of s, equals g of t. There are two common properties that are essential to inverse Laplace transforms. First is the linearity property which states that Laplace inverse of a into capital G1 of s plus b into Capital G2 of s equals a into g1 of t plus b into g2 of t. The second property is shifting property which states that if Laplace inverse of capital G of s equals g of t, then Laplace inverse of capital G of s minus a equals e power at into g of t. Let’s look at some common inverse Laplace forms and their transforms. The Laplace inverse of one upon s is 1. Laplace inverse of one upon s minus a is e power at. For n factorial upon s power n plus one it is t power n, where n is any positive integer greater than zero. Laplace inverse for a upon s squared plus a squared is sin at and for s upon s squared plus a squared is cos at. Finally, the inverse Laplace transform of s upon s squared minus a squared is cosh at and a upon s squared minus a squared is sinh at. Hence, we first saw what Inverse Laplace transforms and their linearity property is and then went on to see the inverse Laplace transform of a few common functions.
In the next episode of MechanicaLEi find out What applications of boundary value problems are?

Attributions:

Doh De Oh by Kevin MacLeod is licensed under a Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/)
Source: http://incompetech.com/music/royalty-free/index.html?isrc=USUAN1100255
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Subtle Library by Fabian Measures (http://freemusicarchive.org/music/Fabian_Measures/) is licensed under a Creative Commons Attribution licence ( https://creativecommons.org/licenses/by/4.0/)
Source: http://freemusicarchive.org/music/Fabian_Measures/Emotion_Music/Subtle_Library