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:

$G(x_{1},x_{2},\dots ,x_{n})=g_{A}({\mbox{only the variables }}x_{i},i\in A)g_{B}({\mbox{only the variables }}x_{i},i\in B)$

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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:

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

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:

$\int _{R}G(x,y)\,dA=\prod _{i=1}^{n}\left(\int _{a_{i}}^{b_{i}}g_{i}(x_{i})\,dx_{i}\right)$