Homogeneous function
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In mathematics, a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor.
For example, a homogeneous function of two variables x and y is a realvalued function that satisfies the condition for some constant and all real numbers . The constant k is called the degree of homogeneity.
More generally, if ƒ : V → W is a function between two vector spaces over a field F, and k is an integer, then ƒ is said to be homogeneous of degree k if

(1)
for all nonzero α ∈ F and v ∈ V. When the vector spaces involved are over the real numbers, a slightly less general form of homogeneity is often used, requiring only that (1) hold for all α > 0.
Homogeneous functions can also be defined for vector spaces with the origin deleted, a fact that is used in the definition of sheaves on projective space in algebraic geometry. More generally, if S ⊂ V is any subset that is invariant under scalar multiplication by elements of the field (a "cone"), then a homogeneous function from S to W can still be defined by (1).
Contents
Examples[edit]
Example 1[edit]
The function is homogeneous of degree 2:
For example, suppose x = 2, y = 4 and t = 5. Then
 , and
 .
Linear functions[edit]
Any linear map ƒ : V → W is homogeneous of degree 1 since by the definition of linearity
for all α ∈ F and v ∈ V. Similarly, any multilinear function ƒ : V_{1} × V_{2} × ... V_{n} → W is homogeneous of degree n since by the definition of multilinearity
for all α ∈ F and v_{1} ∈ V_{1}, v_{2} ∈ V_{2}, ..., v_{n} ∈ V_{n}. It follows that the nth differential of a function ƒ : X → Y between two Banach spaces X and Y is homogeneous of degree n.
Homogeneous polynomials[edit]
Monomials in n variables define homogeneous functions ƒ : F^{n} → F. For example,
is homogeneous of degree 10 since
The degree is the sum of the exponents on the variables; in this example, 10=5+2+3.
A homogeneous polynomial is a polynomial made up of a sum of monomials of the same degree. For example,
is a homogeneous polynomial of degree 5. Homogeneous polynomials also define homogeneous functions.
Given a homogeneous polynomial of degree k, it is possible to get a homogeneous function of degree 1 by raising to the power 1/k. So for example, for every k the following function is homogeneous of degree 1:
Min/max[edit]
For every set of weights , the following functions are homogeneous of degree 1:
Polarization[edit]
A multilinear function g : V × V × ... V → F from the nth Cartesian product of V with itself to the underlying field F gives rise to a homogeneous function ƒ : V → F by evaluating on the diagonal:
The resulting function ƒ is a polynomial on the vector space V.
Conversely, if F has characteristic zero, then given a homogeneous polynomial ƒ of degree n on V, the polarization of ƒ is a multilinear function g : V × V × ... V → F on the nth Cartesian product of V. The polarization is defined by: These two constructions, one of a homogeneous polynomial from a multilinear form and the other of a multilinear form from a homogeneous polynomial, are mutually inverse to one another. In finite dimensions, they establish an isomorphism of graded vector spaces from the symmetric algebra of V^{∗} to the algebra of homogeneous polynomials on V.
Rational functions[edit]
Rational functions formed as the ratio of two homogeneous polynomials are homogeneous functions off of the affine cone cut out by the zero locus of the denominator. Thus, if f is homogeneous of degree m and g is homogeneous of degree n, then f/g is homogeneous of degree m − n away from the zeros of g.
Nonexamples[edit]
Logarithms[edit]
The natural logarithm scales additively and so is not homogeneous.
This can be demonstrated with the following examples: , , and . This is because there is no such that .
Affine functions[edit]
Affine functions (the function is an example) do not scale multiplicatively.
Positive homogeneity[edit]
In the special case of vector spaces over the real numbers, the notion of positive homogeneity often plays a more important role than homogeneity in the above sense. A function ƒ : V \ {0} → R is positively homogeneous of degree k if
for all α > 0. Here k can be any real number. A (nonzero) continuous function homogeneous of degree k on R^{n} \ {0} extends continuously to R^{n} if and only if Re{k} > 0.
Positively homogeneous functions are characterized by Euler's homogeneous function theorem. Suppose that the function ƒ : R^{n} \ {0} → R is continuously differentiable. Then ƒ is positively homogeneous of degree k if and only if
This result follows at once by differentiating both sides of the equation ƒ(αy) = α^{k}ƒ(y) with respect to α, applying the chain rule, and choosing α to be 1. The converse holds by integrating. Specifically, let . Since ,
Thus, . This implies . Therefore, : ƒ is positively homogeneous of degree k.
As a consequence, suppose that ƒ : R^{n} → R is differentiable and homogeneous of degree k. Then its firstorder partial derivatives are homogeneous of degree k − 1. The result follows from Euler's theorem by commuting the operator with the partial derivative.
One can specialise the theorem to the case of a function of a single real variable (n = 1), in which case the function satisfies the ordinary differential equation
 .
This equation may be solved using an integrating factor approach, with solution , where .
Homogeneous distributions[edit]
A continuous function ƒ on R^{n} is homogeneous of degree k if and only if
for all compactly supported test functions ; and nonzero real t. Equivalently, making a change of variable y = tx, ƒ is homogeneous of degree k if and only if
for all t and all test functions ;. The last display makes it possible to define homogeneity of distributions. A distribution S is homogeneous of degree k if
for all nonzero real t and all test functions ;. Here the angle brackets denote the pairing between distributions and test functions, and μ_{t} : R^{n} → R^{n} is the mapping of scalar multiplication by the real number t.
Application to differential equations[edit]
The substitution v = y/x converts the ordinary differential equation
where I and J are homogeneous functions of the same degree, into the separable differential equation
See also[edit]
References[edit]
 Blatter, Christian (1979). "20. Mehrdimensionale Differentialrechnung, Aufgaben, 1.". Analysis II (2nd ed.) (in German). Springer Verlag. p. 188. ISBN 3540094849.
External links[edit]
 Hazewinkel, Michiel, ed. (2001) [1994], "Homogeneous function", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104
 "Homogeneous function". PlanetMath.
 Eric Weisstein. "Euler's Homogeneous Function Theorem". MathWorld.