San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

An Euler-type Theorem for Exponentially
Weighted Homogeneous Functions

Leonhard Euler proved an interesting theorem to the effect that

if F(λx1, λx2, …, λxm) = λnF(x1, x2, …, xm)
Σi=1mxi∂F/∂xi = nF(x1, x2, …, xm)

This can be represented more succinctly by letting X represent the variables {x1, x2, …, xm} as an m-dimensional column vector. A function such that

F(λX) = λnF(X)

is said to be homogeneous of degree n.

Euler's Theorem for Homogeneous Functions is then

(∂F/∂X)·X = nF(X)

where (∂F/∂X) is the partial derivatives of F with respect to the xi represented as a row vector.

Exponentially-Weighted Homogeneous Functions

In physics there arise functions of the form

V(X) = exp(−A·X)F(X)

where F(X) is a homogeneous function and A·X is Σaixi.

For such functions

(∂F/∂X)·X = (n−A·X)V(X)


Differentiate both sides of
V(λX) = exp(−λA·X)λnF(X)
with respect to λ

The result is
(∂V/∂λX)·X = nλn-1exp(−λA·X)F(X) − (A·X)λnexp(−λA·X)F(X)

Now set λ equal to 1 to get
(∂V/∂X)·X = nV(X) − (A·X)V(x) = (n−A·X)V(X)

As an example, consider the Yukawa potential

V(r) = −H*exp(−r/r0)/r

Here m=1, F(r)=−H/r and A=(1/r0). The function F(r) is homogeneous of degree −1.

By the theorem above

r∂V/∂r = (−1 − r/r0)V = −(1+ r/r0)V

HOME PAGE OF applet-magic
HOME PAGE OF Thayer Watkins