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.