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Mathematics
General Math
What is the Definition of the Delta Function?
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[QUOTE="anuttarasammyak, post: 6524654, member: 674023"] There are multiple approaches to delta function. Here I would start from Heaviside Step Funiiton [URL]https://en.wikipedia.org/wiki/Heaviside_step_function[/URL] We can not differentiate Heaviside Step Function at 0 in usual healthy sense. But Dirac boldly invented a strange delta "function" as the result of its derivative including x=0. [tex]H'(x)=\delta(x)[/tex] Say we give 1 Ns impulse to a 1kg body to make it move with 1m/s. There are many ways to do it, e.g. Applying Force 1 N during 1 second, Force 10N during 1/10 second, Force 100N during 1/100second, ----- Force 1/x N during x second, ------ We can make time duration x as small as we like with increasing Force 1/x . There is no limit of x to zero in this sequence because 1/x diverges to infinity. But Dirac say there exists force limit ##\delta(x)##, that is to say, in a instant of time, the infinite force is applied to generate 1 Ns momentum on the body. Delta function is a momentum maker in a instant of time. [tex]\int_{-\infty}^x \delta(\xi) d\xi = H(x)[/tex] Heaviside step function shows thus made momentum by Delta function at x=0. [/QUOTE]
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What is the Definition of the Delta Function?
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