What is Heaviside Unit Step Function?
Channel: MechanicaLEi
Date: 2018-06-29

Welcome back MechanicaLEi, did you know that Heaviside Unit function represents switches or similar functions whose values suddenly change with time? This makes us wonder, what is Heaviside Unit Step Function? Before we jump in check out the previous part of this series to learn about what Linearity Property is? First let see what the first Shifting property of Laplace transform is. Consider, Laplace Transform of f of t is F of s and s is greater than a. Then the first shifting property states that Laplace transform of e power at f of t is equal to F of s minus a. In other words, the substitution s minus a for s in the transform corresponds to the multiplication of the original function by e power at. A Heaviside Unit function is defined as a function say u of t equals o for t less than c and equals q if t greater than c. It can be visualized as a mathematical switch that turns on when the value in the x axis crosses c. The graphs can be any type and not only constants. To deal with the Laplace transforms of such functions, second shifting property is employed. It states that if Laplace transform of f of t is equal to F of s and g is a Heaviside function with g of t equals f of t minus a for t greater than a and g of t equals zero for t less than a. Then Laplace transform of g of t is e power minus as into F of s. So, whatever the value is subtracted from t in the on-switch part of g of t, it simply multiplies by minus t in the power of e.
Hence, We first saw what First shifting property of Laplace Transforms are and then went on to see what Heaviside Unit Function is and how second shifting function is related to it?
In the next episode of MechanicaLEi find out what Dirac Delta function 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