Abel transform of delta function

[rsingh628]

I have a problem on the Abel transform where I tried to use the geometric interpretation definition by substituting for [imath]r[/imath]. How do I compute the integral involving the Dirac delta function below? I thought about changing the variable of integration to something like [imath]r=\sqrt{x^2+y^2}[/imath], but is that the right approach? I’m also having trouble with the integral limits, is it confined to the unit circle? Any help is appreciated.

 

[blamocur]

I thought about changing the variable of integration to something like r=x2+y2r=\sqrt{x^2+y^2}r=x2+y2, but is that the right approach?

This would be my choice too. I don't have much experience in dealing with Dirac's delta function, but the single most useful property is that [imath]\int_{-\infty}^\infty f(t) \delta(t-a) dt = f(a)[/imath] for any "decent" [imath]f(t)[/imath]. This tells me that [imath]r[/imath] would be a more convenient variable for integration. It might also help to remember that [imath]\int_{-\infty}^\infty = \int_{-\infty}^0 + \int_0^\infty[/imath] and that [imath]x[/imath] should be treated as a constant in the right hand side.
Good luck, and keep us posted on your progress.