# Transforming a multivariable integral to make it separable

In the following I will omit requirements of smoothness, extent of domain, finiteness, etc, both to simplify the exposition and because I don't know exactly what the requirements are. Please imagine that such requirements are stated correctly.

Let's say that a function $F : \mathbb{C}^n\to\mathbb{C}$ separates (where $\mathbb{C}$ is the complex numbers) if we can factorize it like $F(z_1,\ldots,z_n)=F_1(z_1)\times\cdots\times F_n(z_n)$. In this case we can factorize the integral of $F$ like $\iiint F(z_1,\ldots,z_n) ~dz_1\cdots dz_n = \int F_1(z_1)dz_1\times \cdots \times\int F_n(z_n)dz_n$.

In case $F$ does not separate, we might be able to change variables to make it separate. If $g : \mathbb{C}^n\to\mathbb{C}^n$ is nice enough and $\Delta$ is its Jacobian determinant (or maybe its inverse depending on which way you like to define it), then $\iiint F(z_1,\ldots,z_n) ~dz_1\cdots dz_n = \iiint G(w_1,\ldots,w_n) ~dw_1\cdots dw_n$, where $G(w_1,\ldots,w_n)=F(g(w_1,\ldots,w_n))\Delta(w_1,\ldots,w_n)^{-1}$.

My question is: for which $F$ can $g$ be chosen so that $G$ separates?

An example everyone knows is $F(z_1,\ldots,z_n)=\exp(Q(z_1,\ldots,z_n))$, where $Q$ is a quadratic form. Then $g$ can be chosen to be a linear transformation that diagonalizes the quadratic form. In general, a non-linear transformation will be required.

• I don't think you can, in general; I'd love to see someone find such a transformation for $\left(\sum_{j=1}^n z_j\right)^m$ Commented Sep 5, 2010 at 9:54