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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a homogeneous function is a
function of several variables In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the functi ...
such that, if all its arguments are multiplied by a
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''degree''; that is, if is an integer, a function of variables is homogeneous of degree if :f(sx_1,\ldots, sx_n)=s^k f(x_1,\ldots, x_n) for every x_1, \ldots, x_n, and s\ne 0. For example, a
homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; t ...
of degree defines a homogeneous function of degree . The above definition extends to functions whose
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...
and
codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either the ...
are
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
: a function f : V \to W between two -vector spaces is ''homogeneous'' of degree k if for all nonzero s \in F and v \in V. This definition is often further generalized to functions whose domain is not , but a
cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines con ...
in , that is, a subset of such that \mathbf\in C implies s\mathbf\in C for every nonzero scalar . In the case of
functions of several real variables In mathematical analysis and its applications, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function o ...
and
real vector space Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
s, a slightly more general form of homogeneity called positive homogeneity is often considered, by requiring only that the above identities hold for s > 0, and allowing any real number as a degree of homogeneity. Every homogeneous real function is ''positively homogeneous''. The converse is not true, but is locally true in the sense that (for integer degrees) the two kinds of homogeneity cannot be distinguished by considering the behavior of a function near a given point. A
norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
over a real vector space is an example of a positively homogeneous function that is not homogeneous. A special case is the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
of real numbers. The quotient of two homogeneous polynomials of the same degree gives an example of a homogeneous function of degree zero. This example is fundamental in the definition of
projective scheme In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...
s.


Definitions

The concept of a homogeneous function was originally introduced for
functions of several real variables In mathematical analysis and its applications, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function o ...
. With the definition of
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s at the end of 19th century, the concept has been naturally extended to functions between vector spaces, since a
tuple In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
of variable values can be considered as a
coordinate vector In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An easy example may be a position such as (5, 2, 1) in a 3-dimensiona ...
. It is this more general point of view that is described in this article. There are two commonly used definitions. The general one works for vector spaces over arbitrary
fields Fields may refer to: Music *Fields (band), an indie rock band formed in 2006 *Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song by ...
, and is restricted to degrees of homogeneity that are
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s. The second one supposes to work over the field of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s, or, more generally, over an
ordered field In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered field ...
. This definition restricts to positive values the scaling factor that occurs in the definition, and is therefore called ''positive homogeneity'', the qualificative ''positive'' being often omitted when there is no risk of confusion. Positive homogeneity leads to consider more functions as homogeneous. For example, the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
and all norms are positively homogeneous functions that are not homogeneous. The restriction of the scaling factor to real positive values allows also considering homogeneous functions whose degree of homogeneity is any real number.


General homogeneity

Let and be two
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
. A
linear cone In linear algebra, a ''cone''—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under scalar multiplication; that is, is a cone if x\in C implies sx\in C for every . W ...
in is a subset of such that sx\in C for all x\in C and all nonzero s\in F. A ''homogeneous function'' from to is a
partial function In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is de ...
from to that has a linear cone as its
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...
, and satisfies :f(sx) = s^kf(x) for some
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
, every x\in C, and every nonzero s\in F. The integer is called the ''degree of homogeneity'', or simply the ''degree'' of . A typical example of a homogeneous function of degree is the function defined by a
homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; t ...
of degree . The
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
defined by the quotient of two homogeneous polynomials is a homogeneous function; its degree is the difference of the degrees of the numerator and the denominator; its ''cone of definition'' is the linear cone of the points where the value of denominator is not zero. Homogeneous functions play a fundamental role in
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pro ...
since any homogeneous function from to defines a well-defined function between the
projectivization In mathematics, projectivization is a procedure which associates with a non-zero vector space ''V'' a projective space (V), whose elements are one-dimensional subspaces of ''V''. More generally, any subset ''S'' of ''V'' closed under scalar multi ...
s of and . The homogeneous rational functions of degree zero (those defined by the quotient of two homogeneous polynomial of the same degre) play an essential role in the
Proj construction In algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective varieties. The construction, while not functo ...
of
projective scheme In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...
s.


Positive homogeneity

When working over the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s, or more generally over an
ordered field In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered field ...
, it is commonly convenient to consider ''positive homogeneity'', the definition being exactly the same as that in the preceding section, with "nonzero " replaced by "" in the definitions of a linear cone and a homogeneous function. This change allow considering (positively) homogeneous functions with any real number as their degrees, since
exponentiation Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
with a positive real base is well defined. Even in the case of integer degrees, there are many useful functions that are positively homogeneous without being homogeneous. This is, in particular, the case of the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
function and norms, which are all positively homogeneous of degree . They are not homogeneous since , -x, =, x, \neq -, x, if x\neq 0. This remains true in the
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
case, since the field of the complex numbers \C and every complex vector space can be considered as real vector spaces.
Euler's homogeneous function theorem In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''deg ...
is a characterization of positively homogeneous
differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
s, which may be considered as the ''fundamental theorem on homogeneous functions''.


Examples


Simple example

The function f(x, y) = x^2 + y^2 is homogeneous of degree 2: f(tx, ty) = (tx)^2 + (ty)^2 = t^2 \left(x^2 + y^2\right) = t^2 f(x, y).


Absolute value and norms

The absolute value of a
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
is a positively homogeneous function of degree , which is not homogeneous, since , sx, =s, x, if s>0, and , sx, =-s, x, if s<0. The absolute value of a
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
is a positively homogeneous function of degree 1 over the real numbers (that is, when considering the complex numbers as a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
over the real numbers). It is not homogeneous, over the real numbers as well as over the complex numbers. More generally, every
norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
and
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...
is a positively homogeneous function of degree which is not a homogeneous function. As for the absolute value, if the norm or semi-norm is defined on a vector space over the complex numbers, this vector space has to be considered as vector space over the real number for applying the definition of a positively homogeneous function.


Linear functions

Any
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
f : V \to W between
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
is homogeneous of degree 1, by the definition of linearity: f(\alpha \mathbf) = \alpha f(\mathbf) for all \alpha \in and v \in V. Similarly, any multilinear function f : V_1 \times V_2 \times \cdots V_n \to W is homogeneous of degree n, by the definition of multilinearity: f\left(\alpha \mathbf_1, \ldots, \alpha \mathbf_n\right) = \alpha^n f(\mathbf_1, \ldots, \mathbf_n) for all \alpha \in and v_1 \in V_1, v_2 \in V_2, \ldots, v_n \in V_n.


Homogeneous polynomials

Monomials In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called power product, is a product of powers of variables with nonnegative integer expone ...
in n variables define homogeneous functions f : \mathbb^n \to \mathbb. For example, f(x, y, z) = x^5 y^2 z^3 \, is homogeneous of degree 10 since f(\alpha x, \alpha y, \alpha z) = (\alpha x)^5(\alpha y)^2(\alpha z)^3 = \alpha^ x^5 y^2 z^3 = \alpha^ f(x, y, z). \, The degree is the sum of the exponents on the variables; in this example, 10 = 5 + 2 + 3. A
homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; t ...
is a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
made up of a sum of monomials of the same degree. For example, x^5 + 2x^3 y^2 + 9xy^4 is a homogeneous polynomial of degree 5. Homogeneous polynomials also define homogeneous functions. Given a homogeneous polynomial of degree k with real coefficients that takes only positive values, one gets a positively homogeneous function of degree k/d by raising it to the power 1 / d. So for example, the following function is positively homogeneous of degree 1 but not homogeneous: \left(x^2 + y^2 + z^2\right)^\frac.


Min/max

For every set of weights w_1,\dots,w_n, the following functions are positively homogeneous of degree 1, but not homogeneous: * \min\left(\frac, \dots, \frac\right) (
Leontief utilities In economics, especially in consumer theory, a Leontief utility function is a function of the form: u(x_1,\ldots,x_m)=\min\left\ . where: * m is the number of different goods in the economy. * x_i (for i\in 1,\dots,m) is the amount of good i in the ...
) * \max\left(\frac, \dots, \frac\right)


Rational functions

Rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
s formed as the ratio of two polynomials are homogeneous functions in their
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...
, that is, off of the
linear cone In linear algebra, a ''cone''—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under scalar multiplication; that is, is a cone if x\in C implies sx\in C for every . W ...
formed by the zeros 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

The homogeneous
real functions In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers \mathbb, or a subset of \mathbb that contains an interva ...
of a single variable have the form x\mapsto cx^k for some constant . So, the
affine function In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, ...
x\mapsto x+5, the
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
x\mapsto \ln(x), and the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
x\mapsto e^x are not homogeneous.


Euler's theorem

Roughly speaking, Euler's homogeneous function theorem asserts that the positively homogeneous functions of a given degree are exactly the solution of a specific
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
. More precisely: ''Proof:'' For having simpler formulas, we set \mathbf x=(x_1, \ldots, x_n). The first part results by using the
chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
for differentiating both sides of the equation f(s\mathbf x ) = s^k f(\mathbf x) with respect to s, and taking the limit of the result when tends to . The converse is proved by integrating a simple
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
. Let \mathbf be in the interior of the domain of . For sufficiently close of , the function g(s) = f(s \mathbf) is well defined. The partial differential equation implies that sg'(s)= k f(s \mathbf)=k g(s). The solutions of this
linear differential equation In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b( ...
have the form g(s)=g(1)s^k. Therefore, f(s \mathbf) = g(s) = s^k g(1) = s^k f(\mathbf), if is sufficiently close to . If this solution of the partial differential equation would not be defined for all positive , then the
functional equation In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
would allow to prolongate the solution, and the partial differential equation implies that this prolongation is unique. So, the domain of a maximal solution of the partial differential equation is a linear cone, and the solution is positively homogeneous of degree . \square As a consequence, if f : \R^n \to \R is continuously differentiable and homogeneous of degree k, its first-order
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
s \partial f/\partial x_i are homogeneous of degree k - 1. This results from Euler's theorem by differentiating the partial differential equation with respect to one variable. In the case of a function of a single real variable (n = 1), the theorem implies that a continuously differentiable and positively homogeneous function of degree has the form f(x)=c_+ x^k for x>0 and f(x)=c_- x^k for x<0. The constants c_+ and c_- are not necessarily the same, as it is the case for the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
.


Application to differential equations

The substitution v = y / x converts the
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
I(x, y)\frac + J(x,y) = 0, where I and J are homogeneous functions of the same degree, into the
separable differential equation In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs ...
x \frac = - \frac - v.


Generalizations


Homogeneity under a monoid action

The definitions given above are all specialized cases of the following more general notion of homogeneity in which X can be any set (rather than a vector space) and the real numbers can be replaced by the more general notion of a
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...
. Let M be a
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...
with identity element 1 \in M, let X and Y be sets, and suppose that on both X and Y there are defined monoid actions of M. Let k be a non-negative integer and let f : X \to Y be a map. Then f is said to be if for every x \in X and m \in M, f(mx) = m^k f(x). If in addition there is a function M \to M, denoted by m \mapsto , m, , called an then f is said to be if for every x \in X and m \in M, f(mx) = , m, ^k f(x). A function is (resp. ) if it is homogeneous of degree 1 over M (resp. absolutely homogeneous of degree 1 over M). More generally, it is possible for the symbols m^k to be defined for m \in M with k being something other than an integer (for example, if M is the real numbers and k is a non-zero real number then m^k is defined even though k is not an integer). If this is the case then f will be called if the same equality holds: f(mx) = m^k f(x) \quad \text x \in X \text m \in M. The notion of being is generalized similarly.


Distributions (generalized functions)

A continuous function f on \R^n is homogeneous of degree k if and only if \int_ f(tx) \varphi(x)\, dx = t^k \int_ f(x)\varphi(x)\, dx for all
compactly supported In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest ...
test function Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives d ...
s \varphi; and nonzero real t. Equivalently, making a
change of variable Change or Changing may refer to: Alteration * Impermanence, a difference in a state of affairs at different points in time * Menopause, also referred to as "the change", the permanent cessation of the menstrual period * Metamorphosis, or change, ...
y = tx, f is homogeneous of degree k if and only if t^\int_ f(y)\varphi\left(\frac\right)\, dy = t^k \int_ f(y)\varphi(y)\, dy for all t and all test functions \varphi. The last display makes it possible to define homogeneity of distributions. A distribution S is homogeneous of degree k if t^ \langle S, \varphi \circ \mu_t \rangle = t^k \langle S, \varphi \rangle for all nonzero real t and all test functions \varphi. Here the angle brackets denote the pairing between distributions and test functions, and \mu_t : \R^n \to \R^n is the mapping of scalar division by the real number t.


Glossary of name variants

Let f : X \to Y be a map between two
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s over a field \mathbb (usually the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s \R or
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s \Complex). If S is a set of scalars, such as \Z, [0, \infty), or \R for example, then f is said to be if f(s x) = s f(x) for every x \in X and scalar s \in S. For instance, every additive map between vector spaces is S := \Q although it Cauchy's functional equation, might not be S := \R. The following commonly encountered special cases and variations of this definition have their own terminology: #() : f(rx) = r f(x) for all x \in X and all real r > 0. #* This property is often also called because for a function valued in a vector space or field, it is
logically equivalent Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
to: f(rx) = r f(x) for all x \in X and all real r \geq 0.Assume that f is strictly positively homogeneous and valued in a vector space or a field. Then f(0) = f(2 \cdot 0) = 2 f(0) so subtracting f(0) from both sides shows that f(0) = 0. Writing r := 0, then for any x \in X, f(r x) = f(0) = 0 = 0 f(x) = r f(x), which shows that f is nonnegative homogeneous. However, for a function valued in the
extended real numbers In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra on ...
\infty, \infty= \R \cup \, which appear in fields like
convex analysis Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory. Convex sets A subset C \subseteq X of s ...
, the multiplication 0 \cdot f(x) will be undefined whenever f(x) = \pm \infty and so these statements are not necessarily interchangeable.However, if such an f satisfies f(rx) = r f(x) for all r > 0 and x \in X, then necessarily f(0) \in \ and whenever f(0), f(x) \in \R are both real then f(r x) = r f(x) will hold for all r \geq 0. #* This property is used in the definition of a
sublinear function In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on a vector space X is a real-valued function with only some of the properties of a seminorm. ...
. #*
Minkowski functional In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space. If K is a subset of a real or complex vector space X, then ...
s are exactly those non-negative extended real-valued functions with this property. #: f(rx) = r f(x) for all x \in X and all real r. #* This property is used in the definition of a
linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the s ...
. #: f(sx) = s f(x) for all x \in X and all scalars s \in \mathbb. #* It is emphasized that this definition depends on the scalar field \mathbb underlying the domain X. #* This property is used in the definition of
linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the s ...
s and
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
s. #: f(sx) = \overline f(x) for all x \in X and all scalars s \in \mathbb. #* If \mathbb = \Complex then \overline typically denotes the
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of s. But more generally, as with
semilinear map In linear algebra, particularly projective geometry, a semilinear map between vector spaces ''V'' and ''W'' over a field ''K'' is a function that is a linear map "up to a twist", hence ''semi''-linear, where "twist" means "field automorphism of ''K' ...
s for example, \overline could be the image of s under some distinguished automorphism of \mathbb. #* Along with
additivity Additive may refer to: Mathematics * Additive function, a function in number theory * Additive map, a function that preserves the addition operation * Additive set-functionn see Sigma additivity * Additive category, a preadditive category with fi ...
, this property is assumed in the definition of an
antilinear map In mathematics, a function f : V \to W between two complex vector spaces is said to be antilinear or conjugate-linear if \begin f(x + y) &= f(x) + f(y) && \qquad \text \\ f(s x) &= \overline f(x) && \qquad \text \\ \end hold for all vectors x, y ...
. It is also assumed that one of the two coordinates of a
sesquilinear form In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows o ...
has this property (such as the
inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
of a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
). All of the above definitions can be generalized by replacing the condition f(rx) = r f(x) with f(rx) = , r, f(x), in which case that definition is prefixed with the word or For example,
  1. : f(sx) = , s, f(x) for all x \in X and all scalars s \in \mathbb. * This property is used in the definition of a
    seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...
    and a
    norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
    .
If k is a fixed real number then the above definitions can be further generalized by replacing the condition f(rx) = r f(x) with f(rx) = r^k f(x) (and similarly, by replacing f(rx) = , r, f(x) with f(rx) = , r, ^k f(x) for conditions using the absolute value, etc.), in which case the homogeneity is said to be (where in particular, all of the above definitions are ). For instance,
  1. : f(rx) = r^k f(x) for all x \in X and all real r.
  2. : f(sx) = s^k f(x) for all x \in X and all scalars s \in \mathbb.
  3. : f(rx) = , r, ^k f(x) for all x \in X and all real r.
  4. : f(sx) = , s, ^k f(x) for all x \in X and all scalars s \in \mathbb.
A nonzero
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
that is homogeneous of degree k on \R^n \backslash \lbrace 0 \rbrace extends continuously to \R^n if and only if k > 0.


See also

*
Homogeneous space In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ' ...
*


Notes

;Proofs


References

* * *


External links

* * {{MathWorld, title=Euler's Homogeneous Function Theorem, urlname=EulersHomogeneousFunctionTheorem, author=Eric Weisstein Linear algebra Differential operators Types of functions Leonhard Euler