Homogeneous function

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This editable Main Article is under development and subject to a disclaimer.

[edit intro]

In mathematics, a function f,

f:ℝn->ℝ

is homogeneous of degree p, if

f(λ𝐱)=λpf(𝐱),𝐱∈ℝn,λ∈ℝ,p∈ℕ*.

The degree of homogeneity p is a positive integral number.

### Examples[edit]

f(x)=axm⟹f(λx)=a(λx)m=λm(axm)=λmf(x)(degreem).f(x,y,z)=x2yz+3xy2z−5xyz2⟹f(λx,λy,λz)=λ4x2yz+3λ4xy2z−5λ4xyz2=λ4f(x,y,z)(degree4)

## Euler's theorem[edit]

Let f be differentiable and homogeneous of order p, then

∑i=1nxi∂f(x1,…,xn)∂xi=pf(x1,…,xn)

### Proof[edit]

By the chain rule

df(λx1,…,λxn)dλ=∑i=1n∂f(λx1,…,λxn)∂(λxi)dλxidλ=∑i=1nxi∂f(λx1,…,λxn)∂(λxi).(1)

From the homogeneity,

df(λx1,…,λxn)dλ=dλpdλf(x1,…,xn)=pλp−1f(x1,…,xn).(2)

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