Multiplicatively separable function

Definition

For a function of two variables

Suppose $G$ is a function of two variables $x$ and $y$. We say that $G$ is multiplicatively separable if there exist functions $f,g$ of one variable such that:

$G(x,y)=f(x)g(y)$

on the entire domain of $G$.

Note that the concept of multiplicatively separable is sensitive to the coordinate system, i.e., if we change the coordinate system, a function that was originally multiplicatively separable need not remain multiplicatively separable.

For a function of many variables

Suppose $G$ is a function of $n$ variables $x_1,x_2,\dots ,x_n$. We say that $G$ is completely multiplicatively separable if there exist functions $g_1,g_2,\dots ,g_n$, each a function of one variable, such that:

$G(x_1,x_2,\dots ,x_n)=g_1(x_1)g_2(x_2)\dots g_n(x_n)$

(note that the subscripts here are not to be confused with subscripts used for partial derivatives).

There is a weaker notion of partially multiplicatively separable: if we express the set $\{1,2,\dots ,n\}$ as a union of two disjoint subsets $A,B$, $G$ is multiplicatively separable with respect to the partition if there exist functions $g_A,g_B$ such that:

$G(x_1,x_2,\dots ,x_n)=g_A(\text{only the variables }{x}_i,i\in A)g_B(\text{only the variables }{x}_i,i\in B)$

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Partial derivatives

For a function of two variables

Consider the case $G(x,y)=f(x)g(y)$.

Then, if $f$ is $m$ times differentiable and $g$ is $n$ times differentiable, then $G_{xx\dots xyy\dots y}$ makes sense where $x$ occurs $m$ times and $y$ occurs $n$ times, and:

$G_{xx\dots xyy\dots y}=f^{(m)}(x)g^{(n)}(y)$

Further, any partial derivative of $G$ that uses $m$ occurrences of $x$ and $n$ occurrences of $y$ will have the same derivative as the above.

In particular, we have that:

• $G_x(x,y)=f^{\prime }(x)g(y)$
• $G_y(x,y)=f(x)g^{\prime }(y)$
• $G_{xy}(x,y)=G_{yx}(x,y)=f^{\prime }(x)g^{\prime }(y)$