Multiplicatively separable function

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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)