Is there a table of Abel transforms for Gaussian distributions available?

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The discussion centers on the availability of a table for Abel transforms specifically for Gaussian distributions. The transformation is defined as f(r) -> f(x): exp(-r²/2σ²) transforms to (2π)¹/²σ*exp(-x²/2σ²). The participant, Dave, expresses confusion regarding the effect of the added constant on the value of sigma and questions the expected narrowing of the distribution when viewed as a line integral.

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Mr Boom
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Hello. I'm wondering if anyone has a table of transforms showing the result of an Abel transform on a Gaussian distribution. I have been unable to find the solution to this. Many thanks for any help. I'm reconstructing an an image from a picture that fits a Gaussian very well, hence I'm hoping to just use an analytical solution.

Dave
 
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and the answer is...

So I found the solution described as f(r) -> f(x):

exp(-r2/2σ2) -> (2π)1/2σ*exp(-x2/2σ2)

The added constant does not change the the value of sigma so, which doesn't make sense to me. When viewing the line itegral, shouldn't the distribution appear more narrow?
 
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