Homogeneous function

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In mathematics, a function f,

${\displaystyle f:{\mathbb{ℝ}}^n\to \mathbb{ℝ}}$

is homogeneous of degree p, if

${\displaystyle f(\lambda \mathbf{𝐱})={\lambda }^{p}f(\mathbf{𝐱}),\mathbf{𝐱}\in {\mathbb{ℝ}}^n,\lambda \in \mathbb{ℝ},p\in {\mathbb{\mathbb{N}}}^{*}.}$

The degree of homogeneity p is a positive integral number.

Examples[edit]

${\displaystyle \begin{array}{rl}f(x)=& a{x}^m⟹f(\lambda x)=a(\lambda x{)}^m={\lambda }^m(a{x}^m)={\lambda }^{m}f(x)(\text{degree}m).\\ f(x,y,z)=& x^{2}yz+3x{y}^{2}z-5xy{z}^2⟹f(\lambda x,\lambda y,\lambda z)={\lambda }^4{x}^{2}yz+3{\lambda }^{4}x{y}^{2}z-5{\lambda }^{4}xy{z}^2={\lambda }^{4}f(x,y,z)(\text{degree}4)\\ \end{array}}$

Euler's theorem[edit]

Let f be differentiable and homogeneous of order p, then

${\displaystyle {\displaystyle \sum _{i=1}^n}{x}_i\frac{\partial f(x_1,\dots ,x_n)}{\partial x_i}=pf(x_1,\dots ,x_n)}$

Proof[edit]

By the chain rule

${\displaystyle \frac{df(\lambda x_1,\dots ,\lambda x_n)}{d\lambda }={\displaystyle \sum _{i=1}^n}\frac{\partial f(\lambda x_1,\dots ,\lambda x_n)}{\partial (\lambda x_i)}\frac{d\lambda x_i}{d\lambda }={\displaystyle \sum _{i=1}^n}{x}_i\frac{\partial f(\lambda x_1,\dots ,\lambda x_n)}{\partial (\lambda x_i)}.(1)}$

From the homogeneity,

${\displaystyle \frac{df(\lambda x_1,\dots ,\lambda x_n)}{d\lambda }=\frac{d{\lambda }^p}{d\lambda }f(x_1,\dots ,x_n)=p{\lambda }^{p-1}f(x_1,\dots ,x_n).(2)}$

Compare Eqs (1) and (2) for λ = 1 and the result to be proved follows.