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An Euler-type Theorem for Exponentially Weighted Homogeneous Functions

Leonhard Euler proved an interesting theorem to the effect that

if F(λx_{1}, λx_{2}, …, λx_{m}) = λ^{n}F(x_{1}, x_{2}, …, x_{m})
then
Σ_{i=1}^{m}x_{i}∂F/∂x_{i} =
nF(x_{1}, x_{2}, …, x_{m})

This can be represented more succinctly by letting X represent the variables {x_{1}, x_{2}, …, x_{m}}
as an m-dimensional column vector. A function such that

F(λX) = λ^{n}F(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 x_{i} 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 Σa_{i}x_{i}.

For such functions

(∂F/∂X)·X = (n−A·X)V(X)

Proof

Differentiate both sides of
V(λX) = exp(−λA·X)λ^{n}F(X)
with respect to λ

The result is
(∂V/∂λX)·X = nλ^{n-1}exp(−λA·X)F(X) − (A·X)λ^{n}exp(−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/r_{0})/r

Here m=1, F(r)=−H/r and A=(1/r_{0}). The function F(r) is homogeneous of degree −1.