Dirac delta function how did they prove this?

[galactic]

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?

 

[HallsofIvy]

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 $\delta(x)$ or $\delta(-x)$ for specific values of x. What is true is that $\delta$ applied to an odd function is 0.