San José State University

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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(λx1, λx2, …, λxm) = λnF(x1, x2, …, xm) then 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

## Proof

#### 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