I'm having some troubles in proving these identities relating two good functions (Blue and Red, always reals, always positive, always continuous together with their arguments always reals, positive) $$\color{blue}{B\left( x \right)} = - \int_x^\infty {\left[ {\int_0^\infty \color{blue}{B (x')}\color{red}{R(y)}{\mkern 1mu} dy - \int_0^{x'} \color{red}{R (y)}\color{blue}{B(x' - y)}{\mkern 1mu} dy} \right]} {\mkern 1mu} dx',$$ and $$\color{blue}{B\left( x \right)} = \int_0^x {\left[ {\int_{x - x'}^\infty \color{blue} {B (x')}\color{red}{R(y)}{\mkern 1mu} dy} \right]} {\mkern 1mu} dx'.$$ ...but putting inside whatever shape for Blue and Red... they give the same color (Blue) so probably I'd say... no typos.
I'm getting "colorblinded" :-/
Thanks JD
This is not an answer. It is just for inserting a figure.
The RHS of the second expression amounts to a sweeping of the domain in form of a rectangular trapezoid as represented on the figure below.
I imagine that the first expression corresponds to an integration on a larger domain to which is substracted a superfluous quantity... but all my attempts have been unsuccessful...
May I ask you to check that all the bounds (in particular) are correct in this first expression ?