I have an integrand consisting of the variables $a,b,w,x,y,z$. Now I integrate over the variables $w,x,y,z$ ($w$ and $x$ from $0$ to $1$ and $y$ and $z$ from $0$ to infinity).
I know that the resulting expression after the integration should be antisymmetric under exchange of the variables $a,b$.
Now my question is: Is already the integrand antisymmetric under exchange of the variables $a,b$? I assume that it should be the case, but unfortunately it's not the case when I check that numerically.
Thanks for your help!
It is not, in general, true that if $F(x,y) = \int f(x,y,\vec z) \,\mathrm{d}\vec z$ is (anti-)symmetric, then $f(x,y, \vec z)$ is (anti-)symmetric for $x, y$. This follows from the fact that one is (in some sense) averaging over the $z$ values.
For example, consider the function $f : \mathbb{R}^2\times [0,1] \to \mathbb R$ given by $f(x,y,z) = x^2z - y^2$. Then $F(x,y) = \int_0^1 f(x,y,z)\,\mathrm{d}z = x^2-y^2$ is anti-symmetric in $x$ and $y$, even though $f(x,y,z)$ is not.