What are Laplace Transforms?

Welcome back MechanicalEI, did you know that Laplace Transformations were bought into popularity after its use during world war 2? This makes us wonder, what are Laplace Transforms? Before we jump in check out the previous part of this series to learn about what Euler's Column Theory is? Now, The Laplace transform of a function f(t), defined for all real numbers t ≥ 0, is the function F(s), which is a unilateral transform defined by F of s equals integral of f of t into e power minus s t with respect to dt from zero to infinity, where s is a complex number frequency parameter and is equal to sigma plus iota omega, with real numbers sigma and omega. An alternate notation for the Laplace transform is L of f instead of F. The meaning of the integral depends on types of functions of interest. A necessary condition for existence of the integral is that f must be locally integrable on closed interval of 0 to infinity. While studying Laplace transforms it is also important to know the definition of Bounded Variables. In mathematical analysis, a function of bounded variation, also known as BV function, is a real-valued function whose total variation is bounded (finite). For a continuous function of a single variable, being of bounded variation means that the distance along the direction of the y-axis, neglecting the contribution of motion along x-axis, traveled by a point moving along the graph has a finite value. Hence, we first saw what Laplace Transforms are and then went on to see what functions of bounded variation are?
In the next episode of MechanicalEI find out what Linearity Property is?
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
Artist: http://incompetech.com/

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

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