# Homogeneous function

of degree $\lambda$

A function $f$ such that for all points $( x _ {1} \dots x _ {n} )$ in its domain of definition and all real $t > 0$, the equation

$$f ( t x _ {1} \dots t x _ {n} ) = \ t ^ \lambda f ( x _ {1} \dots x _ {n} )$$

holds, where $\lambda$ is a real number; here it is assumed that for every point $( x _ {1} \dots x _ {n} )$ in the domain of $f$, the point $( t x _ {1} \dots t x _ {n} )$ also belongs to this domain for any $t > 0$. If

$$f ( x _ {1} \dots x _ {n} ) = \ \sum _ {0 \leq k _ {1} + \dots + k _ {n} \leq m } a _ {k _ {1} \dots k _ {n} } x _ {1} ^ {k _ {1} } \dots x _ {n} ^ {k _ {n} } ,$$

that is, $f$ is a polynomial of degree not exceeding $m$, then $f$ is a homogeneous function of degree $m$ if and only if all the coefficients $a _ {k _ {1} \dots k _ {n} }$ are zero for $k _ {1} + \dots + k _ {n} < m$. The concept of a homogeneous function can be extended to polynomials in $n$ variables over an arbitrary commutative ring with an identity.

Suppose that the domain of definition $E$ of $f$ lies in the first quadrant, $x _ {1} > 0 \dots x _ {n} > 0$, and contains the whole ray $( t x _ {1} \dots t x _ {n} )$, $t > 0$, whenever it contains $( x _ {1} \dots x _ {n} )$. Then $f$ is homogeneous of degree $\lambda$ if and only if there exists a function $\phi$ of $n- 1$ variables, defined on the set of points of the form $( x _ {2} / x _ {1} \dots x _ {n} / x _ {1} )$ where $( x _ {1} \dots x _ {n} ) \in E$, such that for all $( x _ {1} \dots x _ {n} ) \in E$,

$$f ( x _ {1} \dots x _ {n} ) = \ x _ {1} ^ \lambda \phi \left ( { \frac{x _ 2}{x _ 1} } \dots { \frac{x _ n}{x _ 1} } \right ) .$$

If the domain of definition $E$ of $f$ is an open set and $f$ is continuously differentiable on $E$, then the function is homogeneous of degree $\lambda$ if and only if for all $( x _ {1} \dots x _ {n} )$ in its domain of definition it satisfies the Euler formula

$$\sum _ { i=1 } ^ { n } x _ {i} \frac{\partial f ( x _ {1} \dots x _ {n} ) }{\partial x _ {i} } = \ \lambda f ( x _ {1} \dots x _ {n} ) .$$

How to Cite This Entry:
Homogeneous function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homogeneous_function&oldid=51769
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article