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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 real-valued function that satisfies the condition f(αx,αy)=αkf(x,y)displaystyle f(alpha x,alpha y)=alpha ^kf(x,y) for some constant kdisplaystyle k and all real numbers αdisplaystyle alpha . 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

f(αv)=αkf(v)displaystyle f(alpha mathbf v )=alpha ^kf(mathbf v )

(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).

Examples

A homogeneous function is not necessarily continuous, as shown by this example. This is the function f defined by f(x,y)=xdisplaystyle f(x,y)=x if xy>0displaystyle xy>0 or f(x,y)=0displaystyle f(x,y)=0 if xy≤0displaystyle xyleq 0. This function is homogeneous of degree 1, i.e. f(αx,αy)=αf(x,y)displaystyle f(alpha x,alpha y)=alpha f(x,y) for any real numbers α,x,ydisplaystyle alpha ,x,y. It is discontinuous at y=0,x≠0displaystyle y=0,xneq 0.

Example 1

The function f(x,y)=x2+y2displaystyle f(x,y)=x^2+y^2 is homogeneous of degree 2:
f(tx,ty)=(tx)2+(ty)2=t2(x2+y2)=t2f(x,y).displaystyle f(tx,ty)=(tx)^2+(ty)^2=t^2(x^2+y^2)=t^2f(x,y).

For example, suppose x = 2, y = 4 and t = 5. Then

• 
  f(x,y)=22+42=4+16=20displaystyle f(x,y)=2^2+4^2=4+16=20, and


• 
  f(5x,5y)=52(22+42)=25(20)=500displaystyle f(5x,5y)=5^2(2^2+4^2)=25(20)=500.

Linear functions

Any linear map ƒ : V → W is homogeneous of degree 1 since by the definition of linearity

f(αv)=αf(v)displaystyle f(alpha mathbf v )=alpha f(mathbf v )

for all α ∈ F and v ∈ V. Similarly, any multilinear function ƒ : V₁ × V₂ × ... V_n → W is homogeneous of degree n since by the definition of multilinearity

f(αv1,…,αvn)=αnf(v1,…,vn)displaystyle f(alpha mathbf v _1,ldots ,alpha mathbf v _n)=alpha ^nf(mathbf v _1,ldots ,mathbf v _n)

for all α ∈ F and v₁ ∈ V₁, v₂ ∈ V₂, ..., v_n ∈ V_n. It follows that the n-th differential of a function ƒ : X → Y between two Banach spaces X and Y is homogeneous of degree n.

Homogeneous polynomials

Monomials in n variables define homogeneous functions ƒ : Fⁿ → F. For example,

f(x,y,z)=x5y2z3displaystyle f(x,y,z)=x^5y^2z^3,

is homogeneous of degree 10 since

f(αx,αy,αz)=(αx)5(αy)2(αz)3=α10x5y2z3=α10f(x,y,z).displaystyle f(alpha x,alpha y,alpha z)=(alpha x)^5(alpha y)^2(alpha z)^3=alpha ^10x^5y^2z^3=alpha ^10f(x,y,z).,

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,

x5+2x3y2+9xy4displaystyle x^5+2x^3y^2+9xy^4,

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:

(xk+yk+zk)1/kdisplaystyle (x^k+y^k+z^k)^1/k

Min/max

For every set of weights w1,…,wndisplaystyle w_1,dots ,w_n, the following functions are homogeneous of degree 1:

• 
  min(x1/w1,…,xn/wn)displaystyle min(x_1/w_1,dots ,x_n/w_n) (Leontief utilities)


• max(x1/w1,…,xn/wn)displaystyle max(x_1/w_1,dots ,x_n/w_n)

Polarization

A multilinear function g : V × V × ... V → F from the n-th 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:

f(v)=g(v,v,…,v).displaystyle f(v)=g(v,v,dots ,v).

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 n-th Cartesian product of V. The polarization is defined by:
g(v1,v2,…,vn)=1n!∂∂t1∂∂t2⋯∂∂tnf(t1v1+⋯+tnvn).displaystyle g(v_1,v_2,dots ,v_n)=frac 1n!frac partial partial t_1frac partial partial t_2cdots frac partial partial t_nf(t_1v_1+cdots +t_nv_n).
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

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.

Non-examples

Logarithms

The natural logarithm f(x)=ln⁡xdisplaystyle f(x)=ln x scales additively and so is not homogeneous.

This can be demonstrated with the following examples: f(5x)=ln⁡5x=ln⁡5+f(x)displaystyle f(5x)=ln 5x=ln 5+f(x), f(10x)=ln⁡10+f(x)displaystyle f(10x)=ln 10+f(x), and f(15x)=ln⁡15+f(x)displaystyle f(15x)=ln 15+f(x). This is because there is no kdisplaystyle k such that f(α⋅x)=αk⋅f(x)displaystyle f(alpha cdot x)=alpha ^kcdot f(x).

Affine functions

Affine functions (the function f(x)=x+5displaystyle f(x)=x+5 is an example) do not scale multiplicatively.

Positive homogeneity

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

f(αx)=αkf(x)displaystyle f(alpha x)=alpha ^kf(x),

for all α > 0. Here k can be any real number. A (nonzero) continuous function homogeneous of degree k on Rⁿ  0 extends continuously to Rⁿ if and only if Rek > 0.

Positively homogeneous functions are characterized by Euler's homogeneous function theorem. Suppose that the function ƒ : Rⁿ 0 → R is continuously differentiable. Then ƒ is positively homogeneous of degree k if and only if

x⋅∇f(x)=kf(x).displaystyle mathbf x cdot nabla f(mathbf x )=kf(mathbf x ).

This result follows at once by differentiating both sides of the equation ƒ(αy) = α^kƒ(y) with respect to α, applying the chain rule, and choosing α to be 1. The converse holds by integrating. Specifically, let
g(α)=f(αx)displaystyle textstyle g(alpha )=f(alpha mathbf x ).
Since αx⋅∇f(αx)=kf(αx)displaystyle textstyle alpha mathbf x cdot nabla f(alpha mathbf x )=kf(alpha mathbf x ),

g′(α)=x⋅∇f(αx)=kαf(αx)=kαg(α).displaystyle g'(alpha )=mathbf x cdot nabla f(alpha mathbf x )=frac kalpha f(alpha mathbf x )=frac kalpha g(alpha ).

Thus, g′(α)−kαg(α)=0displaystyle textstyle g'(alpha )-frac kalpha g(alpha )=0.
This implies g(α)=g(1)αkdisplaystyle textstyle g(alpha )=g(1)alpha ^k.
Therefore, f(αx)=g(α)=αkg(1)=αkf(x)displaystyle textstyle f(alpha mathbf x )=g(alpha )=alpha ^kg(1)=alpha ^kf(mathbf x ): ƒ is positively homogeneous of degree k.

As a consequence, suppose that ƒ : Rⁿ → R is differentiable and homogeneous of degree k. Then its first-order partial derivatives ∂f/∂xidisplaystyle partial f/partial x_i are homogeneous of degree k − 1. The result follows from Euler's theorem by commuting the operator x⋅∇displaystyle mathbf x cdot nabla 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

f′(x)−kxf(x)=0displaystyle f'(x)-frac kxf(x)=0.

This equation may be solved using an integrating factor approach, with solution f(x)=cxkdisplaystyle textstyle f(x)=cx^k, where c=f(1)displaystyle c=f(1).

Homogeneous distributions

A continuous function ƒ on Rⁿ is homogeneous of degree k if and only if

∫Rnf(tx)φ(x)dx=tk∫Rnf(x)φ(x)dxdisplaystyle int _mathbb R ^nf(tx)varphi (x),dx=t^kint _mathbb R ^nf(x)varphi (x),dx

for all compactly supported test functions φdisplaystyle varphi ; and nonzero real t. Equivalently, making a change of variable y = tx, ƒ is homogeneous of degree k if and only if

t−n∫Rnf(y)φ(y/t)dy=tk∫Rnf(y)φ(y)dydisplaystyle t^-nint _mathbb R ^nf(y)varphi (y/t),dy=t^kint _mathbb R ^nf(y)varphi (y),dy

for all t and all test functions φdisplaystyle varphi ;. The last display makes it possible to define homogeneity of distributions. A distribution S is homogeneous of degree k if

t−n⟨S,φ∘μt⟩=tk⟨S,φ⟩displaystyle t^-nlangle S,varphi circ mu _trangle =t^klangle S,varphi rangle

for all nonzero real t and all test functions φdisplaystyle varphi ;. Here the angle brackets denote the pairing between distributions and test functions, and μ_t : Rⁿ → Rⁿ is the mapping of scalar multiplication by the real number t.

Application to differential equations

Main article: Homogeneous differential equation

The substitution v = y/x converts the ordinary differential equation

I(x,y)dydx+J(x,y)=0,displaystyle I(x,y)frac mathrm d ymathrm d x+J(x,y)=0,

where I and J are homogeneous functions of the same degree, into the separable differential equation

xdvdx=−J(1,v)I(1,v)−v.displaystyle xfrac mathrm d vmathrm d x=-frac J(1,v)I(1,v)-v.

See also

• Weierstrass elliptic function


• Triangle center function


• Production function

References

• 
  Blatter, Christian (1979). "20. Mehrdimensionale Differentialrechnung, Aufgaben, 1.". Analysis II (2nd ed.) (in German). Springer Verlag. p. 188. ISBN 3-540-09484-9..mw-parser-output cite.citationfont-style:inherit.mw-parser-output qquotes:"""""""'""'".mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em
  

External links

• 
  Hazewinkel, Michiel, ed. (2001) [1994], "Homogeneous function", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  


• 
  "Homogeneous function". PlanetMath.
  


• Eric Weisstein. "Euler's Homogeneous Function Theorem". MathWorld.