Definition:Homogeneous Function

Definition

Let $f: V \to W$ be a function between two vector spaces $V$ and $W$ over a field $F$.

Then $f$ is homogeneous of degree $n$ if:

$f \left({\alpha \mathbf v}\right) = \alpha^n f \left({\mathbf v}\right)$

for all nonzero $\mathbf v \in V$ and $\alpha \in F$.

A special case is when $f: \R^2 \to \R$ is a real function of two variables.

Then $f \left({x, y}\right)$ is homogeneous of degree $n$ if:

$\exists n \in \Z: \forall t \in \R: f \left({tx, ty}\right) = t^n f \left({x, y}\right)$

Thus, loosely speaking, a homogeneous function of $x$ and $y$ is one where $x$ and $y$ are both of the same "power".

Another special case is when $n = 0$:

$f \left({\alpha \mathbf v}\right) = \alpha^0 f \left({\mathbf v}\right) = f \left({\mathbf v}\right)$

or:

$f \left({tx, ty}\right) = f \left({x, y}\right)$

This is, of course, called a homogeneous function of degree zero.

Sources

• George F. Simmons: Differential Equations (1972): $\S 7$