Laplace transform of the Dirac delta function | Laplace transform | MechanicaLEi

Welcome back MechanicaLEi, did you know that Dirac delta functions are suited to functions that exert a “large” force over a “small” time frame like a hammer striking an object or a short in an electrical system? This makes us wonder, what is Dirac Delta function? Before we jump in check out the previous part of this series to learn about what Heaviside unit step function?
Now, In the above examples we saw a large force (or voltage) would be exerted on the system over a very short time frame. The Dirac Delta function is used to deal with these kinds of forcing functions, simply because Heaviside Unit functions cannot handle these kinds of large functions. Dirac delta functions can be seen as the first derivative of Heaviside unit functions as speed is for distance. One important property of Dirac delta function is del of t minus a equals zero where t is not equal to a. At t equals a, Dirac function is sometimes thought of as having infinite value. SO, the Laplace transform of del of t minus a is given by integral of e power minus st into del of t minus a dt from zero to infinity equal to e power minus as, provided a is greater than zero. Periodic functions as the name suggests are those functions which repeat after some t; that means, function f of t is said to be a periodic function, if f of t plus capital T is equals to f t and t for all t greater than equal to zero. The Laplace transform of f of t is given by integral of f of t e power minus st dt from zero to capital t upon 1 minus e power minus s into capital T.
Hence, we first saw what Dirac delta function is and then went on to see what periodic functions and their Laplace transforms are?

In the next episode of MechanicaLEi find out what Inverse Laplace transforms?

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