Dirac delta function how did they prove this?

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SUMMARY

The Dirac delta function is a distribution, not a conventional function, and is defined within integrals. It exhibits the property that δ(x) = δ(-x) when integrated over a symmetric interval, such as from -a to b. The proof of this relationship involves a change of variables in the integral, which maintains the equality due to the properties of distributions. Specifically, δ applied to an odd function results in zero, reinforcing the even nature of the Dirac delta function in the context of integration.

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Hi all,

I'm familiar with the fact that the dirac delta function (when defined within an integral is even)

Meaning delta(x)= delta(-x) on the interval -a to b when integral signs are present

I want to prove this this relationship but I don't know how to do it other than with a limit maybe

Book said they proved it using a change of variables and changing limits of integration but I can't see how they proved it? Does anyone know how?
 
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What you have written is pretty much meaningless. The "Dirac Delta function" is not a function at all. It is a "distribution" or "generalized function" which means it is applied to functions, not numbers as are ordinary functions. In particular, that means that we do not define either [itex]\delta(x)[/itex] or [itex]\delta(-x)[/itex] for specific values of x. What is true is that [itex]\delta[/itex] applied to an odd function is 0.
 

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