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| {{inline citations|date=July 2013}}
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| In [[mathematics]], a '''homogeneous function''' is a function with [[multiplication|multiplicative]] scaling behaviour: if the argument is multiplied by a [[Coefficient|factor]], then the result is multiplied by some power of this factor. More precisely, if {{nowrap|''ƒ'' : ''V'' → ''W''}} is a [[function (mathematics)|function]] between two [[vector space]]s over a [[field (mathematics)|field]] ''F'', and ''k'' is an integer, then ''ƒ'' is said to be homogeneous of degree ''k'' if
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| {{NumBlk|:|<math> f(\alpha \mathbf{v}) = \alpha^k f(\mathbf{v}) </math>|{{EquationRef|1}}}}
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| for all nonzero {{nowrap|α ∈ ''F''}} and {{nowrap|'''v''' ∈ ''V''}}. This implies it has [[scale invariance]]. When the vector spaces involved are over the [[real numbers]], a slightly more general form of homogeneity is often used, requiring only that ({{EquationNote|1}}) hold for all α > 0.
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| Homogeneous functions can also be defined for vector spaces with the origin deleted, a fact that is used in the definition of [[sheaf (mathematics)|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 an homogeneous function from ''S'' to ''W'' can still be defined by ({{EquationNote|1}}).
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| ==Examples==
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| [[File:HomogeneousDiscontinuousFunction.gif|thumb|A homogeneous function is not necessarily [[continuous function|continuous]], as shown by this example. This is the function ''f'' defined by <math>f(x,y)=x</math> if <math>xy>0</math> or <math>f(x,y)=0</math> if <math>xy \leq 0</math>. This function is homogeneous of order 1, i.e. <math>f(\alpha x, \alpha y)= \alpha f(x,y)</math> for any real numbers <math>\alpha,x,y</math>. It is discontinuous at <math>y=0</math>.]]
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| ===Linear functions===
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| Any [[linear function]] {{nowrap|''ƒ'' : ''V'' → ''W''}} is homogeneous of degree 1, since by the definition of linearity
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| :<math>f(\alpha \mathbf{v})=\alpha f(\mathbf{v})</math>
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| for all {{nowrap|α ∈ ''F''}} and {{nowrap|'''v''' ∈ ''V''}}. Similarly, any [[multilinear]] function {{nowrap|''ƒ'' : ''V''<sub>1</sub> × ''V''<sub>2</sub> × ... ''V''<sub>''n''</sub> → ''W''}} is homogeneous of degree n, since by the definition of multilinearity
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| :<math>f(\alpha \mathbf{v}_1,\ldots,\alpha \mathbf{v}_n)=\alpha^n f(\mathbf{v}_1,\ldots, \mathbf{v}_n)</math>
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| for all {{nowrap|α ∈ ''F''}} and {{nowrap|'''v'''<sub>1</sub> ∈ ''V''<sub>1</sub>}}, {{nowrap|'''v'''<sub>2</sub> ∈ ''V''<sub>2</sub>}}, ..., {{nowrap|'''v'''<sub>''n''</sub> ∈ ''V''<sub>''n''</sub>}}. It follows that the ''n''-th [[Gâteaux derivative#Higher derivatives|differential]] of a function {{nowrap|''ƒ'' : ''X'' → ''Y''}} between two [[Banach space]]s ''X'' and ''Y'' is homogeneous of degree ''n''.
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| ===Homogeneous polynomials===
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| {{main|Homogeneous polynomial}}
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| [[Monomials]] in ''n'' variables define homogeneous functions {{nowrap|''ƒ'' : ''F''<sup>''n''</sup> → ''F''}}. For example,
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| :<math>f(x,y,z)=x^5y^2z^3 \,</math>
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| is homogeneous of degree 10 since
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| :<math>f(\alpha x, \alpha y, \alpha z) = (\alpha x)^5(\alpha y)^2(\alpha z)^3=\alpha^{10}x^5y^2z^3 = \alpha^{10} f(x,y,z). \,</math>
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| The degree is the sum of the exponents on the variables; in this example, 10=5+2+3.
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| A homogeneous polynomial is a polynomial made up of a sum of monomials of the same degree. For example,
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| :<math>x^5 + 2 x^3 y^2 + 9 x y^4 \,</math>
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| is a homogeneous polynomial of degree 5. Homogeneous polynomials also define homogeneous functions.
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| ===Polarization===
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| A multilinear function {{nowrap|''g'' : ''V'' × ''V'' × ... ''V'' → ''F''}} from the ''n''-th [[Cartesian product]] of ''V'' with itself to the underlying [[Field (mathematics)|field]] ''F'' gives rise to an homogeneous function {{nowrap|''ƒ'' : ''V'' → ''F''}} by evaluating on the diagonal:
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| :<math>f(v) = g(v,v,\dots,v).</math>
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| The resulting function ''ƒ'' is a polynomial on the vector space ''V''.
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| Conversely, if ''F'' has characteristic zero, then given an homogeneous polynomial ''ƒ'' of degree ''n'' on ''V'', the [[polarization of an algebraic form|polarization]] of ''ƒ'' is a multilinear function {{nowrap|''g'' : ''V'' × ''V'' × ... ''V'' → ''F''}} on the ''n''-th Cartesian product of ''V''. The polarization is defined by
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| :<math>g(v_1,v_2,\dots,v_n) = \frac{1}{n!} \frac{\partial}{\partial t_1}\frac{\partial}{\partial t_2}\cdots \frac{\partial}{\partial t_n}f(t_1v_1+\cdots+t_nv_n).</math>
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| These two constructions, one of an homogeneous polynomial from a multilinear form and the other of a multilinear form from an homogeneous polynomial, are mutually inverse to one another. In finite dimensions, they establish an isomorphism of [[graded vector space]]s from the [[symmetric algebra]] of ''V''<sup>∗</sup> to the algebra of homogeneous polynomials on ''V''.
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| ===Rational functions===
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| [[Rational function]]s 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''.
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| ==Non-examples==
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| ===Logarithms===
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| The natural logarithm <math>f(x) = \ln x</math> scales additively and so is not homogeneous.
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| This can be proved by noting that <math>f(5x) = \ln 5x = \ln 5 + f(x)</math>, <math>f(10x) = \ln 10 + f(x)</math>, and <math>f(15x) = \ln 15 + f(x)</math>. Therefore <math> \nexists \; k </math> such that <math>f(\alpha \cdot x) = \alpha^k \cdot f(x)</math>.
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| ===Affine functions===
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| Affine functions (the function <math>f(x) = x + 5</math> is an example) do not scale multiplicatively.
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| ==Positive homogeneity==
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| In the special case of vector spaces over the [[real numbers]], the notation of positive homogeneity often plays a more important role than homogeneity in the above sense. A function {{nowrap|''ƒ'' : ''V'' \ {0} → '''R'''}} is positive homogeneous of degree ''k'' if
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| :<math>f(\alpha x) = \alpha^k f(x) \, </math>
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| for all {{nowrap|α > 0}}. Here ''k'' can be any complex number. A (nonzero) continuous function homogeneous of degree ''k'' on '''R'''<sup>''n''</sup> \ {0} extends continuously to '''R'''<sup>''n''</sup> if and only if {{nowrap|Re{''k''} > 0}}.
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| Positive homogeneous functions are characterized by '''Euler's homogeneous function theorem'''. Suppose that the function {{nowrap|''ƒ'' : '''R'''<sup>''n''</sup> \ {0} → '''R'''}} is [[continuously differentiable]]. Then ''ƒ'' is positive homogeneous of degree ''k'' if and only if
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| :<math> \mathbf{x} \cdot \nabla f(\mathbf{x})= kf(\mathbf{x}).</math>
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| This result follows at once by differentiating both sides of the equation {{nowrap|1=''ƒ''(α'''y''') = α<sup>''k''</sup>''ƒ''('''y''')}} with respect to α, applying the [[chain rule]], and choosing {{nowrap|α}} to be 1. The converse holds by integrating. Specifically, let
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| <math>\textstyle g(\alpha) = f(\alpha \mathbf{x})</math>.
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| Since <math>\textstyle \alpha \mathbf{x} \cdot \nabla f(\alpha \mathbf{x})= k f(\alpha \mathbf{x})</math>,
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| :<math>
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| g'(\alpha)
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| = \mathbf{x} \cdot \nabla f(\alpha \mathbf{x})
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| = \frac{k}{\alpha} f(\alpha \mathbf{x})
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| = \frac{k}{\alpha} g(\alpha).
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| </math>
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| Thus, <math>\textstyle g'(\alpha) - \frac{k}{\alpha} g(\alpha) = 0</math>.
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| This implies <math>\textstyle g(\alpha) = g(1) \alpha^k</math>.
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| Therefore, <math>\textstyle f(\alpha \mathbf{x}) = g(\alpha) = \alpha^k g(1) = \alpha^k f(\mathbf{x})</math>: ''ƒ'' is positive homogeneous of degree ''k''.
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| As a consequence, suppose that {{nowrap|''ƒ'' : '''R'''<sup>''n''</sup> → '''R'''}} is [[differentiable]] and homogeneous of degree ''k''. Then its first-order partial derivatives <math>\partial f/\partial x_i</math> are homogeneous of degree ''k'' − 1. The result follows from Euler's theorem by commuting the operator <math>\mathbf{x}\cdot\nabla</math> with the partial derivative.
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| ==Homogeneous distributions==
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| {{main|Homogeneous distribution}}
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| A [[compact support|compactly supported]] continuous function ƒ on '''R'''<sup>''n''</sup> is homogeneous of degree ''k'' if and only if
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| :<math>\int_{\mathbb{R}^n} f(tx)\varphi(x)\, dx = t^k \int_{\mathbb{R}^n} f(x)\varphi(x)\, dx</math>
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| for all compactly supported [[test function]]s <math>\varphi</math>; and nonzero real ''t''. Equivalently, making a [[integration by substitution|change of variable]] {{nowrap|1=''y'' = ''tx''}}, ƒ is homogeneous of degree ''k'' if and only if
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| :<math>t^{-n}\int_{\mathbb{R}^n} f(y)\varphi(y/t)\, dy = t^k \int_{\mathbb{R}^n} f(y)\varphi(y)\, dy</math> | |
| for all ''t'' and all test functions <math>\varphi</math>;. The last display makes it possible to define homogeneity of [[distribution (mathematics)|distributions]]. A distribution ''S'' is homogeneous of degree ''k'' if
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| :<math>t^{-n}\langle S, \varphi\circ\mu_t\rangle = t^k\langle S,\varphi\rangle</math>
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| for all nonzero real ''t'' and all test functions <math>\varphi</math>;. Here the angle brackets denote the pairing between distributions and test functions, and {{nowrap|μ<sub>''t''</sub> : '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup>}} is the mapping of scalar multiplication by the real number ''t''.
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| ==Application to differential equations==
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| :{{main|Homogeneous differential equation}}
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| The substitution ''v'' = ''y''/''x'' converts the [[ordinary differential equation]]
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| : <math>I(x, y)\frac{\mathrm{d}y}{\mathrm{d}x} + J(x,y) = 0,</math>
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| where ''I'' and ''J'' are homogeneous functions of the same degree, into the [[separable differential equation]]{{dn|date=April 2012}}
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| :<math>x \frac{\mathrm{d}v}{\mathrm{d}x}=-\frac{J(1,v)}{I(1,v)}-v.</math>
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| ==See also==
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| *[[Weierstrass elliptic function]]
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| *[[Triangle center function]]
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| *[[Production function#Homogeneous and homothetic production functions|Production function]]
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| ==References==
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| *{{cite book | author=Blatter, Christian | title=Analysis II (2nd ed.) | publisher=Springer Verlag | year=1979 |language=German |isbn=3-540-09484-9 | pages=188 | chapter=20. Mehrdimensionale Differentialrechnung, Aufgaben, 1.}}
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| ==External links==
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| * {{springer|title=Homogeneous function|id=p/h047670}}
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| * {{planetmath reference|id=6381|title=Homogeneous function}}
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| [[Category:Linear algebra]]
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| [[Category:Differential operators]]
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| [[Category:Types of functions]]
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