the proof to the conjecture

Almost one year ago I already posted a mathematical conjecture that I had came up with while working in some lighting experiments that I was doing. I found the solution by a combination of some strict proof followed by and extrapolation and formula symmetrization, but I didn’t prove it formally. Few days after my post, my friend Bruno Pedrosa Lopez (aka kneda) sent me the formal proof, which makes use of the properties of the Beta and Gamma functions and that of the double factorial. The proof, which I finally post here, is as follows (for a,b even, otherwise the integral cancels out by symmetry):

= = = = = = =

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