# Factoring a multi-variable function into single-variable terms

Suppose I have an arbitrary function $f$ of $n$ variables.

For simplicity, let's assume that $n=3$. Suppose that I have its explicit form, $f(x,y,z)$ and I want to expand it into a sum of terms, each of which is separable in all variables, i.e.

$$f(x,y,z)=\sum_i a_i(x)b_i(y)c_i(z).$$

How would I do that? No additional properties are required and the multiplicative constant can be either factored out, distributed over all 3 functions in each term, or included only in one of the functions, it's irrelevant. What's relevant is that $a_i$,$b_i$ and $c_i$ depend only on a single variable.

I tried using this, but it doesn't work with more complicated functions. In my cases, I need to separate a mess with a lot of $\Gamma$-functions. I don't really care about the form of these single-variable functions, they can be in closed form (if it exists) or they can be expressed through a series expansion, as long as it's exact.

• An example where the referenced link doesn't work would be helpful. – Carl Woll Nov 21 '17 at 19:41
• Anything with a $\Gamma$, even something as simple as $\Gamma(x-y)$ or $\Gamma(xy)$. – PhysSE is Cancer Nov 21 '17 at 20:04
• Are you looking for something like Series[Gamma[x - y], {x, 1, 3}, {y, -1, 3}] or Series[Gamma[x y], {x, 1, 3}, {y, 1, 3}]? – Carl Woll Nov 21 '17 at 20:17
• That doesn't give me the exact solution, it truncates the series, I need the explicit form, something like SeriesCoefficient[Gamma[x y], {x, 1, n}, {y, 1, m}] but this doesn't seem to work for anything more complicated than a polynomial. – PhysSE is Cancer Nov 21 '17 at 20:47
• Might want to check work by Frederick Chapman, Keith Geddes, and one or two others on "Geddes series". But I'm not sure thier approach will recover the exact form as opposed to a tensor product approximation for some region. – Daniel Lichtblau Nov 22 '17 at 16:55