Homogeneous function

A multivariate function f(x)f(\mathbf{x}) is said to be homogeneous of degree α\alpha if for all λ>0\lambda>0 and all points x\mathbf{x} the following equality holds:

f(λx)=λαf(x)f(\lambda \mathbf{x})=\lambda^{\alpha}f(\mathbf{x})

Here, x=(x1,…,xn)∈RN\mathbf{x}=(x_{1},\ldots,x_{n})\in \mathbb{R}^{N} and α\alpha may be any real number, though integers are more common.

Properties

For a homogeneous function of degree α\alpha, the following result, called Euler's homogeneous function theorem, is true:

∑i=1Nxi∂f∂xi=αf\sum_{i=1}^{N} x_{i}\frac{ \partial f }{ \partial x_{i} } =\alpha f

In fact:

0=ddλ(f(λx1,…,λxn)−λαf(x1,…,xn))∣λ=1=(∑i=1Nxi∂f∂xi(λx)−αλα−1f(x))∣λ=1=∑i=1Nxi∂f∂xi(x)−αf(x)\begin{align} 0&=\left.{\frac{d}{d\lambda}(f(\lambda x_{1},\ldots,\lambda x_{n})-\lambda^{\alpha}f(x_{1},\ldots,x_{n}))}\right|_{\lambda=1} \\ &=\left.{\left( \sum_{i=1}^{N}x_{i}\frac{ \partial f }{ \partial x_{i} } (\lambda x)-\alpha \lambda^{\alpha-1}f(x) \right)}\right|_{\lambda=1} \\ &=\sum_{i=1}^{N} x_{i}\frac{ \partial f }{ \partial x_{i} } (x)-\alpha f(x) \end{align}

Examples

An example of a degree 22 homogeneous function is

f(x1,x2,x3)=x1x2+x22+x3x1f(x_{1},x_{2},x_{3})=x_{1}x_{2}+x_{2}^{2}+x_{3}x_{1}