Dirac Delta Function; the Point Function.

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The delta function can be viewed as the derivative of the Heaviside step function,

The main property of the delta function is in the fact that it reaches infinity at a single point and is zero at any other point. Its most important property is that its integral is always one:

∫−∞∞δ(x)dx=1

The δ-function was first introduced by Paul Dirac in the 1930s as part of his pioneering work in the field of quantum mechanics but, the idea may well have been around in mathematical circles for some time before that.

Nevertheless, the δ-function is sometimes referred to as the Dirac δ-function. Clearly, such a function does not exist in the classical (analysis) sense. It was originally referred to by Dirac as an improper function, and he recommended its use in analysis only when it is obvious that no inconsistency will follow from it.

Dirac delta function | Laplace transform | Differential Equations | Khan Academy

https://www.sciencedirect.com/topics/engineering/delta-function

http://mathworld.wolfram.com/DeltaFunction.html

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