(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I would like to prove that [tex]\delta(ax)={\delta(x) \over {|a|}}[/tex].

My problem is that I don't know how the absolute value brackets arise.

2. Relevant equations

[tex]\int_{-\infty}^{\infty} \delta(x)dx = 1[/tex]

3. The attempt at a solution

I start from [tex]\int_{-\infty}^{\infty} \delta (ax) dx[/tex], and make the substitution [tex]u=ax[/tex], so that [tex]du=adx[/tex]. Then I arrive at the answer, but without the absolute value brackets.

I think that it is true that [tex]\delta(ax)=\delta(-ax)=\delta(|a|x)[/tex]. I realize that if a<0, then we would reverse the direction of integration upon the substitution for u, so we would get the negative of the expected result. So do we just manually insert the absolute value brackets to make sure we get the correct sign, or is there a way to get the absolute value brackets to come out in the math?

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# Delta function

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