Definitions

# Homogeneous function

In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor.

## Formal definition

Suppose that $f: V rarr W qquadqquad$ is a function between two vector spaces over a field $F qquadqquad$.

We say that $f qquadqquad$ is homogeneous of degree $k qquadqquad$ if

$f\left(alpha mathbf\left\{v\right\}\right) = alpha^k f\left(mathbf\left\{v\right\}\right)$
for all nonzero $alpha isin F qquadqquad$ and $mathbf\left\{v\right\} isin V qquadqquad$.

## Examples

• A linear function $f: V rarr W qquadqquad$ is homogeneous of degree 1, since by the definition of linearity

$f\left(alpha mathbf\left\{v\right\}\right)=alpha f\left(mathbf\left\{v\right\}\right)$
for all $alpha isin F qquadqquad$ and $mathbf\left\{v\right\} isin V qquadqquad$.

• A multilinear function $f: V_1 times ldots times V_n rarr W qquadqquad$ is homogeneous of degree n, since by the definition of multilinearity

$f\left(alpha mathbf\left\{v\right\}_1,ldots,alpha mathbf\left\{v\right\}_n\right)=alpha^n f\left(mathbf\left\{v\right\}_1,ldots, mathbf\left\{v\right\}_n\right)$
for all $alpha isin F qquadqquad$ and $mathbf\left\{v\right\}_1 isin V_1,ldots,mathbf\left\{v\right\}_n isin V_n qquadqquad$.

• It follows from the previous example that the $n$th Fréchet derivative of a function $f: X rightarrow Y$ between two Banach spaces $X$ and $Y$ is homogeneous of degree $n$.
• Monomials in $n$ real variables define homogeneous functions $f:mathbb\left\{R\right\}^n rarr mathbb\left\{R\right\}$. For example,

$f\left(x,y,z\right)=x^5y^2z^3$
is homogeneous of degree 10 since
$\left(alpha x\right)^5\left(alpha y\right)^2\left(alpha z\right)^3=alpha^\left\{10\right\}x^5y^2z^3$.

$x^5 + 2 x^3 y^2 + 9 x y^4$
is a homogeneous polynomial of degree 5. Homogeneous polynomials also define homogeneous functions.

## Elementary theorems

• Euler's theorem: Suppose that the function $f:mathbb\left\{R\right\}^n rarr mathbb\left\{R\right\}$ is differentiable and homogeneous of degree $k$. Then

$mathbf\left\{x\right\} cdot nabla f\left(mathbf\left\{x\right\}\right)= kf\left(mathbf\left\{x\right\}\right) qquadqquad$.

This result is proved as follows. Writing $f=f\left(x_1,ldots,x_n\right)$ and differentiating the equation

$f\left(alpha mathbf\left\{y\right\}\right)=alpha^k f\left(mathbf\left\{y\right\}\right)$
with respect to $alpha$, we find by the chain rule that
$frac\left\{partial\right\}\left\{partial x_1\right\}f\left(alphamathbf\left\{y\right\}\right)frac\left\{mathrm\left\{d\right\}\right\}\left\{mathrm\left\{d\right\}alpha\right\}\left(alpha y_1\right)+ cdots$
frac{partial}{partial x_n}f(alphamathbf{y})frac{mathrm{d}}{mathrm{d}alpha}(alpha y_n) = k alpha ^{k-1} f(mathbf{y}), so that
$y_1frac\left\{partial\right\}\left\{partial x_1\right\}f\left(alphamathbf\left\{y\right\}\right)+ cdots$
y_nfrac{partial}{partial x_n}f(alphamathbf{y}) = k alpha^{k-1} f(mathbf{y}). The above equation can be written in the del notation as
$mathbf\left\{y\right\} cdot nabla f\left(alpha mathbf\left\{y\right\}\right) = k alpha^\left\{k-1\right\}f\left(mathbf\left\{y\right\}\right), qquadqquad nabla=\left(frac\left\{partial\right\}\left\{partial x_1\right\},ldots,frac\left\{partial\right\}\left\{partial x_n\right\}\right)$,
from which the stated result is obtained by setting $alpha=1$.

• Suppose that $f:mathbb\left\{R\right\}^n rarr mathbb\left\{R\right\}$ is differentiable and homogeneous of degree $k$. Then its first-order partial derivatives $partial f/partial x_i$ are homogeneous of degree $k-1 qquadqquad$.

This result is proved in the same way as Euler's theorem. Writing $f=f\left(x_1,ldots,x_n\right)$ and differentiating the equation

$f\left(alpha mathbf\left\{y\right\}\right)=alpha^k f\left(mathbf\left\{y\right\}\right)$
with respect to $y_i$, we find by the chain rule that
$frac\left\{partial\right\}\left\{partial x_i\right\}f\left(alphamathbf\left\{y\right\}\right)frac\left\{mathrm\left\{d\right\}\right\}\left\{mathrm\left\{d\right\}y_i\right\}\left(alpha y_i\right) = alpha ^k frac\left\{partial\right\}\left\{partial x_i\right\}f\left(mathbf\left\{y\right\}\right)frac\left\{mathrm\left\{d\right\}\right\}\left\{mathrm\left\{d\right\}y_i\right\}\left(y_i\right)$,
so that
$alphafrac\left\{partial\right\}\left\{partial x_i\right\}f\left(alphamathbf\left\{y\right\}\right) = alpha ^k frac\left\{partial\right\}\left\{partial x_i\right\}f\left(mathbf\left\{y\right\}\right)$
and hence
$frac\left\{partial\right\}\left\{partial x_i\right\}f\left(alphamathbf\left\{y\right\}\right) = alpha ^\left\{k-1\right\} frac\left\{partial\right\}\left\{partial x_i\right\}f\left(mathbf\left\{y\right\}\right)$.

## Application to ODEs

The substitution $v=y/x$ converts the ordinary differential equation

$I\left(x, y\right)frac\left\{mathrm\left\{d\right\}y\right\}\left\{mathrm\left\{d\right\}x\right\} + J\left(x,y\right) = 0,$
where $I$ and $J$ are homogeneous functions of the same degree, into the separable differential equation
$x frac\left\{mathrm\left\{d\right\}v\right\}\left\{mathrm\left\{d\right\}x\right\}=-frac\left\{J\left(1,v\right)\right\}\left\{I\left(1,v\right)\right\}-v$.

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