Selberg Integral - Change of Variables I am trying to evaluate a Selberg-Like integral and I need to make the integral separable (unbinding the variables), but I can't come up with an appropriate change of variables. The integral is:$$\underset{[0,1]^m}{\int\dots\int} d^2z_1\dots d^2z_m \prod_{i<j} |z_i-z_j|^{-\gamma^2} f(z_1,\dots, z_m).$$
I am thinking about this as a function of $\gamma$, so the function $f(z_1,\dots, z_m)$ doesn't really matter yet. I need to introduce a change of variables such that $\prod_{i<j} |z_i-z_)|^{-\gamma^2}$ can be split into terms that include only one variable each. The case with $m=2$ is easy (polar coordinates). For $m=3$, I tried the following: $x_1-x_2=y_1$ and $x_1-x_3=y_1y_2$. This works fine when $m=3$, but I am not sure it can be extended to any $m\in\mathbb{N}$.
 A: If you just want to split the $z_{i}-z_{j}$ and don't care if you complicate the $f(z_{1},\cdots ,z_{n})$, then the Fourier representation is the way to go. To simplify, I will sketch the method using real variables x, the extension to complex variables is tedious, but I think the principle holds.
Start from $\prod_{i<j}g^{2}(|x_{i}-x_{j}|)=\prod_{i\neq j}g(|x_{i}-x_{j}|)$,
Define $g(x)$such that $g(x)=|x|^{-\frac{1}{2}\gamma^{2}}\text{for }x\in[0,1]$ and $0$ elsewhere.
Define $g=\int dke^{ikx}\tilde{g}(k)$ (the Fourier representation) and suppose you can evaluate it in closed form (exercise for the reader).
The Integral $$ I=\int dx_{1}\cdots dx_{n}\prod_{i\neq j}g(x_{i}-x_{j})f(x_{1},\cdots,x_{n})$$ becomes $$I =\int dx_{1}\cdots dx_{n}\int \prod_{i\neq j} dk_{ij}\tilde{g}(k_{ij})e^{i\sum_{i\neq j}k_{ij}(-x_{i}+x_{j})}f(x_{1},\cdots,x_{n})$$
let us examine the exponent: $\sum_{i\neq j}k_{ij}(-x_{i}+x_{j})=\sum_{i}x_{i}\sum_{j}(k_{ji}-k_{ij})$
So,
$$I=\int dx_{1}\cdots dx_{n}\int \prod_{i\neq j} dk_{ij}\tilde{g}(k_{ij})e^{i\sum x_{i}\sum_{j}(k_{ji}-k_{ij})}f(x_{1}\cdots x_{n}).$$ Notice that the x integration is actually a Fourier transform of $f$ (at least if we extend the function $f$ to be zero outside the $x$ integration range.
$$I=\int\prod_{ij} dk_{ij}\tilde{g}(k_{ij})\tilde{f}(\sum_{p}(k_{p1}-k_{1p}),\cdots,\sum_{p}(k_{pn}-k_{np}))$$ which is as far as I can get without making more assumptions.
Whether this helps, or just moves the difficulty elsewhere is up to you.
