Homogeneous function
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In mathematics, a homogeneous function is a function with nice scaling behaviour: if the argument is multiplied by some factor, then the result is multiplied by some power of this factor.
Formally, let f : V → W be a function between two vector spaces over a field F. We say that f is homogeneous of degree k if the equation
- <math> f(\alpha v) = \alpha^k f(v) \qquad\qquad (*) <math>
holds for all α ∈ F and v ∈ V.
A function f(x) = f(x1, ..., xn) that is homogeneous of degree k has partial derivatives of degree k − 1. Furthermore, it satisfies Euler's homogeneous function theorem, which states that x · ∇f(x) = kf(x). Written out in components, this is
- <math>
\sum_{i=1}^n x_i \frac{\partial f}{\partial x_i} (x) = k f(x). <math>
More generally, a function f is said to be homogeneous if the equation f(αv) = g(α) f(v) holds for some strictly increasing positive function g.
Sometimes, a function satisfying (*) for all positive α is said to be positively homogeneous (this requires the field F to be the reals).