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:

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

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:

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

For a function of many variables

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

Suppose $m_1,m_2,\dots,m_n$ are (possibly equal, possibly distinct) nonnegative integers such that each $f_i$ is $m_i$ times differentiable. Now, consider a partial derivative of $G$ that involves $m_1$ differentiations in $x_1$, $m_2$ differentiations in $x_2$, and so on, with $m_i$ differentiations in each $x_i$. The order of the differentiations does not matter. Then, this partial derivative equals:

$\! g_1^{(m_1)}(x_1)g_2^{(m_2)}(x_2) \dots g_n^{(m_n)}(x_n)$

Integration on rectangular regions

For a function of two variables

Suppose $G(x,y) = f(x)g(y)$ is a function of two variables. Consider a rectangular region $R$ of the form $[a,b] \times [p,q]$ where $a,b,p,q$ are numbers. Then:

$\int_R G(x,y) \, dA = \left(\int_a^b f(x) \, dx\right) \left(\int_p^q g(y) \, dy \right)$

For a function of many variables

Suppose $G(x_1,x_2,\dots,x_n) = g_1(x_1)g_2(x_2) \dots g_n(x_n)$ is a function of $n$ variables. Consider a rectangular region $R$ of the form $[a_1,b_1] \times [a_2,b_2] \times \dots \times [a_n,b_n]$. Then:

$\int_R G(x,y) \, dA = \prod_{i=1}^n \left(\int_{a_i}^{b_i} g_i(x_i) \, dx_i \right)$