Let $x, y ,z$ be real numbers and let $a>0.$ Denote $t := x-y-z$ and $$F(x,y, z,t) = \frac{1}{\sqrt{x^2 + a}+\sqrt{y^2 + a}+\sqrt{z^2 + a}-\sqrt{t^2 + a}}. $$
I'd like to know whether we can find $f, g, h , k$ such that
$$F(x,y, z,t) = f(x) g(y) h(z) k(t).$$
$f$ might be constant !
Thank you for any hint.
If $F(x,y,z,t)$ could be written as $F(x,y,z,t)=f(x)g(y)h(z)k(t)$, then $$\frac{F(x,0,2y,x-2y)}{F(x,y,y,x-2y)}=\frac{g(0)h(2y)}{g(y)h(2y)}$$ in particular, this would not depend on $x$. You can write out the fraction but the $x$ do not cancel out. Hence you cannot find such function $f,g,h,k$.