Abel transform of delta function

rsingh628

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May 31, 2021
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I have a problem on the Abel transform where I tried to use the geometric interpretation definition by substituting for rr. How do I compute the integral involving the Dirac delta function below? I thought about changing the variable of integration to something like r=x2+y2r=\sqrt{x^2+y^2}, 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.

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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 f(t)δ(ta)dt=f(a)\int_{-\infty}^\infty f(t) \delta(t-a) dt = f(a) for any "decent" f(t)f(t). This tells me that rr would be a more convenient variable for integration. It might also help to remember that =0+0\int_{-\infty}^\infty = \int_{-\infty}^0 + \int_0^\infty and that xx should be treated as a constant in the right hand side.
Good luck, and keep us posted on your progress.
 
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