Definition:Homogeneous Function/Real Space

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This page is about Homogeneous Real Function. For other uses, see Homogeneous.

Definition

Let $f: \R^2 \to \R$ be a real-valued function of two variables.

$\map f {x, y}$ is a homogeneous function of degree zero if and only if:

$\exists n \in \Z: \forall t \in \R: \map f {t x, t y} = t^n \map f {x, y}$


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


Degree

The integer $n$ is known as the degree of $f$.


Zero Degree

A special case is when $n = 0$:


$f \left({x, y}\right)$ is a homogeneous function of degree zero or of zero degree if and only if:

$\forall t \in \R: f \left({t x, t y}\right) = t^0 f \left({x, y}\right) = f \left({x, y}\right)$


Sources