📂Functions

Homogeneous Function

Definition

For a constant $a$ and a function $f$, if there exists a $k \in \mathbb{N}$ that satisfies the following conditions, then $f$ is called a $k$-th degree homogeneous function.

$$f(ax) = a^{k}f(x)$$

In the case of a multivariable function,

$$f(ax_{1}, ax_{2}, \dots, ax_{n}) = a^{k}f(x_{1}, x_{2}, \dots, x_{n})$$

Explanation

In the case of a univariate function, it is similar to a polynomial function that only contains the highest order term. For example, a second degree homogeneous function is a quadratic function that only contains the quadratic term. If $f(x) = x^{2}$ then,

$$f(ax) = a^{2}x^{2} = a^{2}f(x)$$

For a multivariable function, the sum of the degrees of all variables in each term must be the same. For example, for the bivariate function $f(x,y)$ to be a homogeneous function, it must be of the following form:

$$f(x,y) = ax^{2} + bxy + cy^{2}$$

If a term like $x^{2}y$ is included, it does not satisfy the definition of a homogeneous function.