Homogeneous function

Homogeneous Degree Function $q$ ${\ displaystyle q}$ $q$ - numerical function $f:\mathbb {R} ^{n}\to \mathbb {R}$ ${\ displaystyle f: \ mathbb {R} ^ {n} \ to \ mathbb {R}}$ ${\ displaystyle f: \ mathbb {R} ^ {n} \ to \ mathbb {R}}$ such that for any $\mathbf {v} \in \mathbb {R} ^{n}$ ${\ displaystyle \ mathbf {v} \ in \ mathbb {R} ^ {n}}$ ${\ displaystyle \ mathbf {v} \ in \ mathbb {R} ^ {n}}$ from function definition area $f$ ${\ displaystyle f}$ ${\ displaystyle f}$ and any $\lambda \in \mathbb {R}$ ${\ displaystyle \ lambda \ in \ mathbb {R}}$ ${\ displaystyle \ lambda \ in \ mathbb {R}}$ equality holds:

f(\lambda \mathbf {v} )=\lambda ^{q}f(\mathbf {v} ).\qquad \qquad (*)

{\ displaystyle f (\ lambda \ mathbf {v}) = \ lambda ^ {q} f (\ mathbf {v}). \ qquad \ qquad (*)}

{\ displaystyle f (\ lambda \ mathbf {v}) = \ lambda ^ {q} f (\ mathbf {v}). \ qquad \ qquad (*)}

Parameter $q$ ${\ displaystyle q}$ $q$ called the order of homogeneity . It is understood that if $\mathbf {v} \in \mathbb {R} ^{n}$ ${\ displaystyle \ mathbf {v} \ in \ mathbb {R} ^ {n}}$ ${\ displaystyle \ mathbf {v} \ in \ mathbb {R} ^ {n}}$ enters the domain of function definition, then all points of the form $\lambda \mathbf {v}$ ${\ displaystyle \ lambda \ mathbf {v}}$ ${\ displaystyle \ lambda \ mathbf {v}}$ also fall within the scope of the function.

Distinguish also

positively homogeneous functions for which the equality $(*)$ ${\ displaystyle (*)}$ $(*)$ performed only for positive $\lambda$ ${\ displaystyle \ lambda}$ $\ lambda$ $(\lambda >0),$ ${\ displaystyle (\ lambda> 0),}$ ${\ displaystyle (\ lambda> 0),}$
absolutely homogeneous functions for which the equality holds
$f(\lambda \mathbf {v} )=|\lambda |^{q}f(\mathbf {v} ),$ ${\ displaystyle f (\ lambda \ mathbf {v}) = | \ lambda | ^ {q} f (\ mathbf {v}),}$ ${\ displaystyle f (\ lambda \ mathbf {v}) = | \ lambda | ^ {q} f (\ mathbf {v}),}$
boundedly homogeneous functions for which the equality $(*)$ ${\ displaystyle (*)}$ $(*)$ only holds for some selected values $\lambda ,$ ${\ displaystyle \ lambda,}$ ${\ displaystyle \ lambda,}$
complex homogeneous functions $f:\mathbb {C} ^{n}\to \mathbb {C}$ ${\ displaystyle f: \ mathbb {C} ^ {n} \ to \ mathbb {C}}$ ${\ displaystyle f: \ mathbb {C} ^ {n} \ to \ mathbb {C}}$ for which equality $(*)$ ${\ displaystyle (*)}$ $(*)$ valid for $\mathbf {v} \in \mathbb {C} ^{n}$ ${\ displaystyle \ mathbf {v} \ in \ mathbb {C} ^ {n}}$ ${\ displaystyle \ mathbf {v} \ in \ mathbb {C} ^ {n}}$ and $\lambda \in \mathbb {R}$ ${\ displaystyle \ lambda \ in \ mathbb {R}}$ ${\ displaystyle \ lambda \ in \ mathbb {R}}$ or $\lambda \in \mathbb {C}$ ${\ displaystyle \ lambda \ in \ mathbb {C}}$ ${\ displaystyle \ lambda \ in \ mathbb {C}}$ (as well as for complex indicators $q\in \mathbb {C}$ ${\ displaystyle q \ in \ mathbb {C}}$ ${\ displaystyle q \ in \ mathbb {C}}$ )

Alternative definition of a homogeneous function

In some mathematical sources, functions that are a solution of a functional equation are called homogeneous

f(\lambda \mathbf {v} )=g(\lambda )f(\mathbf {v} )

with predefined function

g(\lambda )

{\ displaystyle g (\ lambda)}

g(\lambda )

and only then it is proved that

g(\lambda )=\lambda ^{q}.

{\ displaystyle g (\ lambda) = \ lambda ^ {q}.}

g(\lambda )=\lambda ^{q}.

For uniqueness of solution

g(\lambda )=\lambda ^{q}

{\ displaystyle g (\ lambda) = \ lambda ^ {q}}

g(\lambda )=\lambda ^{q}

need an additional condition that the function

f(\mathbf {v} )

{\ displaystyle f (\ mathbf {v})}

f(\mathbf {v} )

is not identically zero and that function

g(\lambda )

{\ displaystyle g (\ lambda)}

g(\lambda )

belongs to a certain class of functions (for example, it was continuous or was monotonous). However, if the function

f(\mathbf {v} )

{\ displaystyle f (\ mathbf {v})}

f(\mathbf {v} )

continuous at least at one point with a nonzero value of the function, then

g(\lambda )

{\ displaystyle g (\ lambda)}

g(\lambda )

must be a continuous function for all values

\lambda ,

{\ displaystyle \ lambda,}

\lambda ,

and thus for a wide class of functions

f(\mathbf {v} )

{\ displaystyle f (\ mathbf {v})}

f(\mathbf {v} )

happening

g(\lambda )\equiv \lambda ^{q}

{\ displaystyle g (\ lambda) \ equiv \ lambda ^ {q}}

g(\lambda )\equiv \lambda ^{q}

- the only one possible.

Justification:

A function identically equal to zero satisfies the functional equation $f(\lambda \mathbf {v} )=g(\lambda )f(\mathbf {v} )$ ${\ displaystyle f (\ lambda \ mathbf {v}) = g (\ lambda) f (\ mathbf {v})}$ $f(\lambda \mathbf {v} )=g(\lambda )f(\mathbf {v} )$ at any choice of function $g(\lambda ),$ ${\ displaystyle g (\ lambda),}$ however, this degenerate case is not of particular interest.

If at some point $\mathbf {v} _{0}$ ${\ displaystyle \ mathbf {v} _ {0}}$ value $f(\mathbf {v} _{0})\neq 0,$ ${\ displaystyle f (\ mathbf {v} _ {0}) \ neq 0,}$ then:

$g(\lambda _{1}\lambda _{2})f(\mathbf {v} _{0})=f(\lambda _{1}\lambda _{2}\mathbf {v} _{0})=g(\lambda _{1})f(\lambda _{2}\mathbf {v} _{0})=g(\lambda _{1})g(\lambda _{2})f(\mathbf {v} _{0})$ ${\ displaystyle g (\ lambda _ {1} \ lambda _ {2}) f (\ mathbf {v} _ {0}) = f (\ lambda _ {1} \ lambda _ {2} \ mathbf {v} _ {0}) = g (\ lambda _ {1}) f (\ lambda _ {2} \ mathbf {v} _ {0}) = g (\ lambda _ {1}) g (\ lambda _ {2 }) f (\ mathbf {v} _ {0})}$ from where: $\forall \lambda _{1},\lambda _{2}:g(\lambda _{1}\lambda _{2})=g(\lambda _{1})g(\lambda _{2});$
$g(\lambda _{1}\lambda _{2})=g(\lambda _{1})g(\lambda _{2})\Leftrightarrow G(\mu _{1}+\mu _{2})=G(\mu _{1})+G(\mu _{2}),$ ${\ displaystyle g (\ lambda _ {1} \ lambda _ {2}) = g (\ lambda _ {1}) g (\ lambda _ {2}) \ Leftrightarrow G (\ mu _ {1} + \ mu _ {2}) = G (\ mu _ {1}) + G (\ mu _ {2}),}$ Where $\mu =\log \lambda ,G(\mu )=\log g(\exp(\mu )).$ ${\ displaystyle \ mu = \ log \ lambda, G (\ mu) = \ log g (\ exp (\ mu)).}$

The functional Cauchy equation $G(\mu _{1}+\mu _{2})=G(\mu _{1})+G(\mu _{2})$ ${\ displaystyle G (\ mu _ {1} + \ mu _ {2}) = G (\ mu _ {1}) + G (\ mu _ {2})}$ has a solution in the form of a linear function: $G(t)=q\cdot t,$ ${\ displaystyle G (t) = q \ cdot t,}$ and for the class of continuous or the class of monotone functions this solution is unique. Therefore, if it is known that $g(\lambda )$ ${\ displaystyle g (\ lambda)}$ continuous or monotonic function then $g(\lambda )\equiv \lambda ^{q}.$ ${\ displaystyle g (\ lambda) \ equiv \ lambda ^ {q}.}$

The proof of the uniqueness of the solution of the functional Cauchy equation

1. With rational

t=m/n

{\ displaystyle t = m / n}

rightly

G(t\mu )=t\cdot G(\mu ),

{\ displaystyle G (t \ mu) = t \ cdot G (\ mu),}

as:

but)

G(2\mu )=G(\mu +\mu )=G(\mu )+G(\mu )=2G(\mu ),

{\ displaystyle G (2 \ mu) = G (\ mu + \ mu) = G (\ mu) + G (\ mu) = 2G (\ mu),}

i.e

G(2\mu )=2G(\mu );

{\ displaystyle G (2 \ mu) = 2G (\ mu);}

b)

2G(\mu /2)=G(\mu /2)+G(\mu /2)=G(\mu /2+\mu /2)=G(\mu ),

{\ displaystyle 2G (\ mu / 2) = G (\ mu / 2) + G (\ mu / 2) = G (\ mu / 2 + \ mu / 2) = G (\ mu),}

i.e

G(\mu /2)=(1/2)G(\mu );

{\ displaystyle G (\ mu / 2) = (1/2) G (\ mu);}

etc.;

2. Since irrational numbers that can be arbitrarily closely “clamped” between two rational numbers, for continuous or monotone functions, the relation

G(t\mu )=t\cdot G(\mu )

{\ displaystyle G (t \ mu) = t \ cdot G (\ mu)}

should also be done for irrational

t;

{\ displaystyle t;}

3. The last step: in the ratio

G(t\mu )=t\cdot G(\mu )

{\ displaystyle G (t \ mu) = t \ cdot G (\ mu)}

should ask

\mu =1.

{\ displaystyle \ mu = 1.}

Note: for wider classes of functions, the considered functional equation may have other, very exotic solutions (see the article “Hamel's Basis” ).

Proof of continuity

g(\lambda ),

${\ displaystyle g (\ lambda),}$ if

f(\mathbf {v} )

{\ displaystyle f (\ mathbf {v})}

continuous at least at one point

Let the function $f(\mathbf {v} )$ ${\ displaystyle f (\ mathbf {v})}$ continuous at a fixed point $\mathbf {v} _{0},$ ${\ displaystyle \ mathbf {v} _ {0},}$ moreover $f(\mathbf {v} _{0})\neq 0.$ ${\ displaystyle f (\ mathbf {v} _ {0}) \ neq 0.}$ Consider the identity

f((1+\varepsilon )\lambda _{0}\mathbf {v} _{0})=g((1+\varepsilon )\lambda _{0})f(\mathbf {v} _{0})=g(\lambda _{0})f((1+\varepsilon )\mathbf {v} _{0}).

{\ displaystyle f ((1+ \ varepsilon) \ lambda _ {0} \ mathbf {v} _ {0}) = g ((1+ \ varepsilon) \ lambda _ {0}) f (\ mathbf {v} _ {0}) = g (\ lambda _ {0}) f ((1+ \ varepsilon) \ mathbf {v} _ {0}).}

At $\varepsilon \to 0$ ${\ displaystyle \ varepsilon \ to 0}$ value $f((1+\varepsilon )\mathbf {v} _{0})$ ${\ displaystyle f ((1+ \ varepsilon) \ mathbf {v} _ {0})}$ committed to $f(\mathbf {v} _{0})$ ${\ displaystyle f (\ mathbf {v} _ {0})}$ due to the continuity of the function $f(\mathbf {v} )$ ${\ displaystyle f (\ mathbf {v})}$ at the point $\mathbf {v} _{0}.$ ${\ displaystyle \ mathbf {v} _ {0}.}$ Insofar as $f(\mathbf {v} _{0})\neq 0,$ ${\ displaystyle f (\ mathbf {v} _ {0}) \ neq 0,}$ then that means $g((1+\varepsilon )\lambda _{0})$ ${\ displaystyle g ((1+ \ varepsilon) \ lambda _ {0})}$ committed to $g(\lambda _{0}),$ ${\ displaystyle g (\ lambda _ {0}),}$ that is, that function $g(\lambda )$ ${\ displaystyle g (\ lambda)}$ continuous at point $\lambda _{0}.$ ${\ displaystyle \ lambda _ {0}.}$ Insofar as $\lambda _{0}$ ${\ displaystyle \ lambda _ {0}}$ can be chosen by any $g(\lambda )$ ${\ displaystyle g (\ lambda)}$ continuous at all points.

Corollary: If a homogeneous function $f(\mathbf {v} )$ ${\ displaystyle f (\ mathbf {v})}$ continuous at point $\mathbf {v} _{0},$ ${\ displaystyle \ mathbf {v} _ {0},}$ then $f(\mathbf {v} )$ ${\ displaystyle f (\ mathbf {v})}$ will be continuous also at all points of the form $\lambda \mathbf {v} _{0}$ ${\ displaystyle \ lambda \ mathbf {v} _ {0}}$ (including when $f(\mathbf {v} _{0})=0$ ${\ displaystyle f (\ mathbf {v} _ {0}) = 0}$ )

Properties

If $f_{1},f_{2},\dots$ ${\ displaystyle f_ {1}, f_ {2}, \ dots}$ - homogeneous functions of the same order $q,$ ${\ displaystyle q,}$ then their linear combination with constant coefficients will be a homogeneous function of the same order $q .$ ${\ displaystyle q.}$
If $f_{1},f_{2},\dots$ ${\ displaystyle f_ {1}, f_ {2}, \ dots}$ - homogeneous functions with orders $q_{1},q_{2},\dots ,$ ${\ displaystyle q_ {1}, q_ {2}, \ dots,}$ then their product will be a homogeneous function with order $q=q_{1}+q_{2}+\dots .$ ${\ displaystyle q = q_ {1} + q_ {2} + \ dots.}$
If $f$ ${\ displaystyle f}$ - homogeneous order function $q,$ ${\ displaystyle q,}$ then her $m$ ${\ displaystyle m}$ degree (not necessarily integer) if it makes sense (i.e. if $m$ ${\ displaystyle m}$ Is an integer, or if the value $f$ ${\ displaystyle f}$ positive) will be a homogeneous order function $m q$ ${\ displaystyle mq}$ on the corresponding field of definition. In particular, if $f$ ${\ displaystyle f}$ - homogeneous order function $q$ ${\ displaystyle q}$ then $1/f$ ${\ displaystyle 1 / f}$ will be a homogeneous order function $(-q)$ ${\ displaystyle (-q)}$ and the scope at the points where $f$ ${\ displaystyle f}$ defined and not equal to zero.
If $f\left(x_{1},x_{2},\dots ,x_{n}\right)$ ${\ displaystyle f \ left (x_ {1}, x_ {2}, \ dots, x_ {n} \ right)}$ - homogeneous order function $p,$ ${\ displaystyle p,}$ but $h_{k}\left(y_{1},y_{2},\dots ,y_{m}\right)$ ${\ displaystyle h_ {k} \ left (y_ {1}, y_ {2}, \ dots, y_ {m} \ right)}$ - homogeneous order functions $q,$ ${\ displaystyle q,}$ then a superposition of functions $F\left(y_{1},y_{2},\dots ,y_{m}\right)=f\left(h_{1},h_{2},\dots ,h_{n}\right)$ ${\ displaystyle F \ left (y_ {1}, y_ {2}, \ dots, y_ {m} \ right) = f \ left (h_ {1}, h_ {2}, \ dots, h_ {n} \ right)}$ will be a homogeneous order function $p q .$ ${\ displaystyle pq.}$
If $f\left(x_{1},x_{2},\dots ,x_{n}\right)$ ${\ displaystyle f \ left (x_ {1}, x_ {2}, \ dots, x_ {n} \ right)}$ - homogeneous function $n$ ${\ displaystyle n}$ degree variables $p,$ ${\ displaystyle p,}$ and hyperplane $x_{1}=x_{2}=\dots =x_{j}=0$ ${\ displaystyle x_ {1} = x_ {2} = \ dots = x_ {j} = 0}$ belongs to its domain of definition, then the function $\left(n-j\right)$ ${\ displaystyle \ left (nj \ right)}$ variables $g\left(x_{j+1},x_{j+2},\dots ,x_{n}\right)=f\left(0,\dots ,0,x_{j+1},\dots ,x_{n}\right)$ ${\ displaystyle g \ left (x_ {j + 1}, x_ {j + 2}, \ dots, x_ {n} \ right) = f \ left (0, \ dots, 0, x_ {j + 1}, \ dots, x_ {n} \ right)}$ is a homogeneous degree function $p .$ ${\ displaystyle p.}$
The logarithm of a homogeneous function of zero order or the logarithm of the module of a homogeneous function of zero order is a homogeneous function of zero order. The logarithm of a homogeneous function or the logarithm of the module of a homogeneous function is a homogeneous function if and only if the homogeneity of the function itself is zero.
The homogeneous function module or absolute homogeneous function module is an absolutely homogeneous function. The module of a homogeneous function or the module of a positively homogeneous function is a positively homogeneous function. The modulus of a homogeneous function of zero order is a homogeneous function of zero order. An absolutely homogeneous zero-order function is a homogeneous zero-order function, and vice versa.
An arbitrary function of a homogeneous function of zero order is a homogeneous function of zero order.
If $h_{k}\left(x_{1},x_{2},\dots ,x_{n}\right)$ ${\ displaystyle h_ {k} \ left (x_ {1}, x_ {2}, \ dots, x_ {n} \ right)}$ —— positively homogeneous order functions $p,$ ${\ displaystyle p,}$ Where $p\neq 0,$ ${\ displaystyle p \ neq 0,}$ but $f\left(x_{1},x_{2},\dots ,x_{n}\right)=g\left(h_{1},h_{2},\dots ,h_{m}\right)$ ${\ displaystyle f \ left (x_ {1}, x_ {2}, \ dots, x_ {n} \ right) = g \ left (h_ {1}, h_ {2}, \ dots, h_ {m} \ right)}$ —— positively homogeneous order function $q,$ ${\ displaystyle q,}$ then function $g\left(y_{1},y_{2},\dots ,y_{m}\right)$ ${\ displaystyle g \ left (y_ {1}, y_ {2}, \ dots, y_ {m} \ right)}$ will be a positively homogeneous order function $q/p$ ${\ displaystyle q / p}$ at all points $y$ ${\ displaystyle y}$ in which the system of equations $y_{1}=h_{1}\left(x_{1},x_{2},\dots ,x_{n}\right)$ ${\ displaystyle y_ {1} = h_ {1} \ left (x_ {1}, x_ {2}, \ dots, x_ {n} \ right)}$ , ..., $y_{m}=h_{m}\left(x_{1},x_{2},\dots ,x_{n}\right)$ ${\ displaystyle y_ {m} = h_ {m} \ left (x_ {1}, x_ {2}, \ dots, x_ {n} \ right)}$ has a solution. If at the same time $p$ ${\ displaystyle p}$ —— an odd integer, then positive uniformity can be replaced with ordinary uniformity. Corollary: if there is a continuous or monotonous function $g(y)$ ${\ displaystyle g (y)}$ , and $g\left(f\left(x_{1},x_{2},\dots ,x_{n}\right)\right)$ ${\ displaystyle g \ left (f \ left (x_ {1}, x_ {2}, \ dots, x_ {n} \ right) \ right)}$ —— homogeneous or positively homogeneous function, where $f\left(x_{1},x_{2},\dots ,x_{n}\right)$ ${\ displaystyle f \ left (x_ {1}, x_ {2}, \ dots, x_ {n} \ right)}$ —— homogeneous or positively homogeneous function of nonzero order, then $g(y)=cy^{m}$ ${\ displaystyle g (y) = cy ^ {m}}$ —— power function at all points $y$ ${\ displaystyle y}$ in which the equation $y=f\left(x_{1},x_{2},\dots ,x_{n}\right)$ ${\ displaystyle y = f \ left (x_ {1}, x_ {2}, \ dots, x_ {n} \ right)}$ has a solution. In particular, $f(x)=cx^{q}$ ${\ displaystyle f (x) = cx ^ {q}}$ —— the only monotonic or continuous function of one variable, which is a homogeneous order function $q$ ${\ displaystyle q}$ . (The proof duplicates the arguments from the section “Alternative Definition of a Homogeneous Function” in this article. Moreover, if we remove the restriction $g(y)$ ${\ displaystyle g (y)}$ —— continuous or monotonous, there may be other, very exotic solutions for $g(y)$ ${\ displaystyle g (y)}$ , see the article “Hamel Basis” .)
If the function $f$ ${\ displaystyle f}$ is a polynomial from $n$ ${\ displaystyle n}$ variables, then it will be a homogeneous degree function $q$ ${\ displaystyle q}$ if and only if $f$ ${\ displaystyle f}$ - homogeneous polynomial of degree $q .$ ${\ displaystyle q.}$ In particular, in this case the order of homogeneity $q$ ${\ displaystyle q}$ must be a natural number or zero. (For proof, we need to group together the monomials of the polynomial $cx_{1}^{i_{1}}x_{2}^{i_{2}}\cdots x_{n}^{i_{n}}$ ${\ displaystyle cx_ {1} ^ {i_ {1}} x_ {2} ^ {i_ {2}} \ cdots x_ {n} ^ {i_ {n}}}$ with the same order of homogeneity $k_{j}=i_{1}+i_{2}+\dots +i_{n}$ ${\ displaystyle k_ {j} = i_ {1} + i_ {2} + \ dots + i_ {n}}$ , substitute the result into equality $(*)$ ${\ displaystyle (*)}$ and use the fact that power functions $\lambda ^{k_{1}},\lambda ^{k_{2}},\dots$ ${\ displaystyle \ lambda ^ {k_ {1}}, \ lambda ^ {k_ {2}}, \ dots}$ with different exponents, including non-integer ones, are linearly independent.) The statement can be generalized to the case of linear combinations of monomials of the form $cx_{1}^{i_{1}}x_{2}^{i_{2}}\cdots x_{n}^{i_{n}}$ ${\ displaystyle cx_ {1} ^ {i_ {1}} x_ {2} ^ {i_ {2}} \ cdots x_ {n} ^ {i_ {n}}}$ with integer indices.
If the final product of polynomials is a homogeneous function, then each factor is a homogeneous polynomial . (For proof, we choose monomials in each factor $cx_{1}^{i_{1}}x_{2}^{i_{2}}\cdots x_{n}^{i_{n}}$ ${\ displaystyle cx_ {1} ^ {i_ {1}} x_ {2} ^ {i_ {2}} \ cdots x_ {n} ^ {i_ {n}}}$ with minimum and maximum homogeneity orders $k=i_{1}+i_{2}+\dots +i_{n}$ ${\ displaystyle k = i_ {1} + i_ {2} + \ dots + i_ {n}}$ . Since after multiplication the resulting polynomial should consist of monomials with the same order of homogeneity, then for each factor the minimum and maximum order of homogeneity must be the same number.) The statement can be generalized to the case of linear combinations of monomials of the form $cx_{1}^{i_{1}}x_{2}^{i_{2}}\cdots x_{n}^{i_{n}}$ ${\ displaystyle cx_ {1} ^ {i_ {1}} x_ {2} ^ {i_ {2}} \ cdots x_ {n} ^ {i_ {n}}}$ with integer indices.
If the numerator and denominator of a fractional rational function $f={\frac {P_{n}(x_{1},\dots ,x_{n})}{Q_{m}(x_{1},\dots ,x_{m})}}$ ${\ displaystyle f = {\ frac {P_ {n} (x_ {1}, \ dots, x_ {n})} {Q_ {m} (x_ {1}, \ dots, x_ {m})}}}$ are homogeneous polynomials , the function will be homogeneous with a uniformity order equal to the difference in the uniformity orders of the numerator and denominator. If a fractional rational function is homogeneous, its numerator and denominator, up to a common factor, are homogeneous polynomials . The statement can be generalized to the case of a fractional rational relation of linear combinations of monomials of the form $cx_{1}^{i_{1}}x_{2}^{i_{2}}\cdots x_{n}^{i_{n}}$ ${\ displaystyle cx_ {1} ^ {i_ {1}} x_ {2} ^ {i_ {2}} \ cdots x_ {n} ^ {i_ {n}}}$ with integer indices.
A homogeneous function of nonzero degree at zero is zero if it is defined there: $f(\mathbf {0} )=0.$ ${\ displaystyle f (\ mathbf {0}) = 0.}$ (It turns out when substituting in equality $(*)$ ${\ displaystyle (*)}$ values $\lambda =0$ ${\ displaystyle \ lambda = 0}$ or, in the case of a negative degree of homogeneity, the values $\mathbf {v} =0.$ ${\ displaystyle \ mathbf {v} = 0.}$ ) A homogeneous function of degree zero, if it is defined at zero, can take any value at this point.
If a homogeneous function of degree zero is continuous at zero, then it is a constant (arbitrary). If a homogeneous function of a negative degree is continuous at zero, then it is the identity zero. (Conversion $\mathbf {v'} =\lambda \mathbf {v}$ ${\ displaystyle \ mathbf {v '} = \ lambda \ mathbf {v}}$ can any point $\mathbf {v}$ ${\ displaystyle \ mathbf {v}}$ arbitrarily close to zero. Therefore, if the function at zero is continuous, then we can express the value of the function at the point $\mathbf {v}$ ${\ displaystyle \ mathbf {v}}$ through its value at the point $\mathbf {0}$ ${\ displaystyle \ mathbf {0}}$ using the relation $\lim _{\lambda \to 0}\lambda ^{q}f(\mathbf {v} )=f(\mathbf {0} ).$ ${\ displaystyle \ lim _ {\ lambda \ to 0} \ lambda ^ {q} f (\ mathbf {v}) = f (\ mathbf {0}).}$ )
A homogeneous function of a positive degree at zero tends to zero in any direction that falls within its domain of definition, and a homogeneous function of a negative degree tends to infinity, the sign of which depends on the direction, unless the function is the identity zero along this direction. A homogeneous function of a positive degree is continuous at zero or can be extended to continuous at zero if its domain of definition is $\varepsilon$ ${\ displaystyle \ varepsilon}$ - neighborhood of zero. A homogeneous function of degree zero can be either discontinuous or continuous at zero, and in the case of discontinuity, it is a constant depending on the direction along each ray with a vertex at the origin, if the direction enters its domain of definition. (It turns out when substituting in equality $(*)$ ${\ displaystyle (*)}$ values $\lambda \to 0.$ ${\ displaystyle \ lambda \ to 0.}$ )
If the homogeneous function $f$ ${\ displaystyle f}$ zero is analytic (that is, it decomposes into a converging Taylor series with a non-zero radius of convergence), then it is a polynomial ( homogeneous polynomial ). In particular, in this case, the homogeneity order must be a natural number or zero. (For the proof, it is enough to represent the function in the form of a Taylor series , group together the members of the Taylor series $cx_{1}^{i_{1}}x_{2}^{i_{2}}\cdots x_{n}^{i_{n}}$ ${\ displaystyle cx_ {1} ^ {i_ {1}} x_ {2} ^ {i_ {2}} \ cdots x_ {n} ^ {i_ {n}}}$ with the same order of homogeneity $k_{j}=i_{1}+i_{2}+\dots +i_{n}$ ${\ displaystyle k_ {j} = i_ {1} + i_ {2} + \ dots + i_ {n}}$ , substitute the result into equality $(*)$ ${\ displaystyle (*)}$ and use that power functions $\lambda ^{k_{1}},\lambda ^{k_{2}},\dots$ ${\ displaystyle \ lambda ^ {k_ {1}}, \ lambda ^ {k_ {2}}, \ dots}$ with different exponents, including non-integer ones, are linearly independent.)
Function $f(x_{1},x_{2},...,x_{n})=x_{1}^{q}\cdot h(x_{2}/x_{1},x_{3}/x_{1},...,x_{n}/x_{1})$ ${\ displaystyle f (x_ {1}, x_ {2}, ..., x_ {n}) = x_ {1} ^ {q} \ cdot h (x_ {2} / x_ {1}, x_ {3 } / x_ {1}, ..., x_ {n} / x_ {1})}$ where $h(t_{2},t_{3},...,t_{n})$ ${\ displaystyle h (t_ {2}, t_ {3}, ..., t_ {n})}$ - function $(n-1)$ ${\ displaystyle (n-1)}$ variables, is a homogeneous function with a uniformity order $q .$ ${\ displaystyle q.}$ Function $f(x_{1},x_{2},...,x_{n})=|x|^{q}\cdot h(x_{2}/x_{1},x_{3}/x_{1},...,x_{n}/x_{1}),$ ${\ displaystyle f (x_ {1}, x_ {2}, ..., x_ {n}) = | x | ^ {q} \ cdot h (x_ {2} / x_ {1}, x_ {3} /x_{1►,...,x_{n►/x_{1}),}$ Where $h(t_{2},t_{3},...,t_{n})$ ${\ displaystyle h (t_ {2}, t_ {3}, ..., t_ {n})}$ - function $(n-1)$ ${\ displaystyle (n-1)}$ variables, is an absolutely homogeneous function with a uniformity order $q .$ ${\ displaystyle q.}$
Euler relation : for differentiable homogeneous functions, the scalar product of their gradient by the vector of its variables is proportional to the function itself with a coefficient equal to the order of homogeneity: $\mathbf {v} \cdot \nabla f(\mathbf {v} )=qf(\mathbf {v} )$ ${\ displaystyle \ mathbf {v} \ cdot \ nabla f (\ mathbf {v}) = qf (\ mathbf {v})}$ or, in the equivalent notation, $\sum x_{k}f'_{x_{k}}=qf.$ ${\ displaystyle \ sum x_ {k} f '_ {x_ {k}} = qf.}$ It turns out at differentiation of equality $(*)$ ${\ displaystyle (*)}$ by $\lambda$ ${\ displaystyle \ lambda}$ at $\lambda =1.$ ${\ displaystyle \ lambda = 1.}$
If $f(x_{1},x_{2},...,x_{n})$ ${\ displaystyle f (x_ {1}, x_ {2}, ..., x_ {n})}$ - differentiable homogeneous function with homogeneity order $q$ ${\ displaystyle q}$ , then its first partial derivatives with respect to each of the independent variables $f'_{x_{k}}(x_{1},x_{2},...,x_{n})$ ${\ displaystyle f '_ {x_ {k}} (x_ {1}, x_ {2}, ..., x_ {n})}$ Are homogeneous functions with the order of homogeneity $q-1$ ${\ displaystyle q-1}$ . To prove it, it suffices to differentiate by $x_{k}$ ${\ displaystyle x_ {k}}$ right and left sides of the identity $f(\lambda x_{1},\lambda x_{2},\ldots ,\lambda x_{n})=\lambda ^{q}f(x_{1},x_{2},\ldots ,x_{n})$ ${\ displaystyle f (\ lambda x_ {1}, \ lambda x_ {2}, \ ldots, \ lambda x_ {n}) = \ lambda ^ {q} f (x_ {1}, x_ {2}, \ ldots , x_ {n})}$ and get the identity $f'_{x_{k}}(\lambda x_{1},\lambda x_{2},\ldots ,\lambda x_{n})=\lambda ^{q-1}f'_{x_{k}}(x_{1},x_{2},\ldots ,x_{n}).$ ${\ displaystyle f '_ {x_ {k}} (\ lambda x_ {1}, \ lambda x_ {2}, \ ldots, \ lambda x_ {n}) = \ lambda ^ {q-1} f' _ { x_ {k}} (x_ {1}, x_ {2}, \ ldots, x_ {n}).}$
If $f(x_{1},x_{2},...,x_{n})$ ${\ displaystyle f (x_ {1}, x_ {2}, ..., x_ {n})}$ Is a homogeneous function with the order of homogeneity $q$ ${\ displaystyle q}$ , then its integral (subject to the existence of such an integral) over any independent variable starting from zero $F(x_{1},x_{2},...,x_{n})=\int _{0}^{x_{1}}f(t,x_{2},...,x_{n})dt$ ${\ displaystyle F (x_ {1}, x_ {2}, ..., x_ {n}) = \ int _ {0} ^ {x_ {1}} f (t, x_ {2}, ... , x_ {n}) dt}$ Are homogeneous functions with the order of homogeneity $q+1.$ ${\ displaystyle q + 1.}$ Evidence: $F(\lambda x_{1},\lambda x_{2},...,\lambda x_{n})=$ ${\ displaystyle F (\ lambda x_ {1}, \ lambda x_ {2}, ..., \ lambda x_ {n}) =}$ $\int _{0}^{\lambda x_{1}}f(t,\lambda x_{2},...,\lambda x_{n})dt=$ ${\ displaystyle \ int _ {0} ^ {\ lambda x_ {1}} f (t, \ lambda x_ {2}, ..., \ lambda x_ {n}) dt =}$ $\lambda \int _{0}^{x_{1}}f(\lambda t',\lambda x_{2},...,\lambda x_{n})dt'=$ ${\ displaystyle \ lambda \ int _ {0} ^ {x_ {1}} f (\ lambda t ', \ lambda x_ {2}, ..., \ lambda x_ {n}) dt' =}$ $\lambda ^{q+1}\int _{0}^{x_{1}}f(t',x_{2},...,x_{n})dt'=$ ${\ displaystyle \ lambda ^ {q + 1} \ int _ {0} ^ {x_ {1}} f (t ', x_ {2}, ..., x_ {n}) dt' =}$ $\lambda ^{q+1}F(x_{1},x_{2},...,x_{n})$ {\ displaystyle \ lambda ^ {q + 1} F (x_ {1}, x_ {2}, ..., x_ {n})} (here the integration variable is replaced $t=\lambda t'$ ${\ displaystyle t = \ lambda t '}$ )
If $f(x_{1},x_{2},...,x_{n})$ ${\ displaystyle f (x_ {1}, x_ {2}, ..., x_ {n})}$ Is a homogeneous function with the order of homogeneity $q$ ${\ displaystyle q}$ , then its fractional derivative ( differential integral ) of order $\alpha$ ${\ displaystyle \ alpha}$ calculated as $G(x_{1},x_{2},...,x_{n})={\frac {1}{\Gamma (n-\alpha )}}{\frac {d^{n}}{dx_{1}^{n}}}\int _{0}^{x_{1}}(x_{1}-t)^{n-\alpha -1}f(t,x_{2},...,x_{n})\,dt$ ${\ displaystyle G (x_ {1}, x_ {2}, ..., x_ {n}) = {\ frac {1} {\ Gamma (n- \ alpha)}} {\ frac {d ^ {n }} {dx_ {1} ^ {n}}} \ int _ {0} ^ {x_ {1}} (x_ {1} -t) ^ {n- \ alpha -1} f (t, x_ {2 }, ..., x_ {n}) \, dt}$ in any independent variable starting from zero (provided that the corresponding integral exists, for which it is necessary to choose $n>\alpha$ ${\ displaystyle n> \ alpha}$ ) Are homogeneous functions with the order of homogeneity $q-\alpha .$ ${\ displaystyle q- \ alpha.}$ Consider the function $H(x_{1},x_{2},...,x_{n})=\int _{0}^{x_{1}}(x_{1}-t)^{n-\alpha -1}f(t,x_{2},...,x_{n})\,dt$ ${\ displaystyle H (x_ {1}, x_ {2}, ..., x_ {n}) = \ int _ {0} ^ {x_ {1}} (x_ {1} -t) ^ {n- \ alpha -1} f (t, x_ {2}, ..., x_ {n}) \, dt}$ . Then $H(\lambda x_{1},\lambda x_{2},...,\lambda x_{n})=$ ${\ displaystyle H (\ lambda x_ {1}, \ lambda x_ {2}, ..., \ lambda x_ {n}) =}$ $\int _{0}^{\lambda x_{1}}(\lambda x_{1}-t)^{n-\alpha -1}f(t,\lambda x_{2},...,\lambda x_{n})\,dt=$ ${\ displaystyle \ int _ {0} ^ {\ lambda x_ {1}} (\ lambda x_ {1} -t) ^ {n- \ alpha -1} f (t, \ lambda x_ {2}, .. ., \ lambda x_ {n}) \, dt =}$ $\lambda \int _{0}^{x_{1}}(\lambda x_{1}-\lambda t')^{n-\alpha -1}f(\lambda t',\lambda x_{2},...,\lambda x_{n})\,dt'=$ ${\ displaystyle \ lambda \ int _ {0} ^ {x_ {1}} (\ lambda x_ {1} - \ lambda t ') ^ {n- \ alpha -1} f (\ lambda t', \ lambda x_ {2}, ..., \ lambda x_ {n}) \, dt '=}$ $\lambda ^{q+n-\alpha }\int _{0}^{x_{1}}(x_{1}-t')^{n-\alpha -1}f(t',x_{2},...,x_{n})dt'=$ ${\ displaystyle \ lambda ^ {q + n- \ alpha} \ int _ {0} ^ {x_ {1}} (x_ {1} -t ') ^ {n- \ alpha -1} f (t', x_ {2}, ..., x_ {n}) dt '=}$ $\lambda ^{q+n-\alpha }H(x_{1},x_{2},...,x_{n})$ ${\ displaystyle \ lambda ^ {q + n- \ alpha} H (x_ {1}, x_ {2}, ..., x_ {n})}$ (here the integration variable is replaced $t=\lambda t'$ ${\ displaystyle t = \ lambda t '}$ ) After $n$ ${\ displaystyle n}$ differentiation by a variable $x_{1}$ ${\ displaystyle x_ {1}}$ homogeneous function $H(x_{1},x_{2},...,x_{n})$ ${\ displaystyle H (x_ {1}, x_ {2}, ..., x_ {n})}$ of order $q+n-\alpha$ ${\ displaystyle q + n- \ alpha}$ becomes a homogeneous function with the order of homogeneity $q-\alpha$ ${\ displaystyle q- \ alpha}$ .
If $f(x_{1},x_{2},...,x_{n})$ ${\ displaystyle f (x_ {1}, x_ {2}, ..., x_ {n})}$ Is a homogeneous function with the order of homogeneity $q$ ${\ displaystyle q}$ then her $n$ ${\ displaystyle n}$ -dimensional convolution with a generalized Abelian kernel, calculated as $H(x_{1},x_{2},...,x_{n})=\int _{0}^{x_{1}}\dots \int _{0}^{x_{n}}(x_{1}^{k_{1}}-t_{1}^{k_{1}})^{\left(\mu _{1}-1\right)/k_{1}}\dots (x_{n}^{k_{n}}-t_{n}^{k_{n}})^{\left(\mu _{n}-1\right)/k_{n}}f(t_{1},...,t_{n})\,dt_{1}\dots dt_{n}$ ${\ displaystyle H (x_ {1}, x_ {2}, ..., x_ {n}) = \ int _ {0} ^ {x_ {1}} \ dots \ int _ {0} ^ {x_ { n}} (x_ {1} ^ {k_ {1}} - t_ {1} ^ {k_ {1}}) ^ {\ left (\ mu _ {1} -1 \ right) / k_ {1}} \ dots (x_ {n} ^ {k_ {n}} - t_ {n} ^ {k_ {n}}) ^ {\ left (\ mu _ {n} -1 \ right) / k_ {n}} f (t_ {1}, ..., t_ {n}) \, dt_ {1} \ dots dt_ {n}}$ (subject to the existence of the corresponding integral) is a homogeneous function with the order of homogeneity $q+\mu _{1}+\dots +\mu _{n}$ ${\ displaystyle q + \ mu _ {1} + \ dots + \ mu _ {n}}$ . Evidence: $H(\lambda x_{1},\lambda x_{2},...,\lambda x_{n})=$ ${\ displaystyle H (\ lambda x_ {1}, \ lambda x_ {2}, ..., \ lambda x_ {n}) =}$ $\int _{0}^{\lambda x_{1}}\dots \int _{0}^{\lambda x_{n}}(\lambda ^{k_{1}}x_{1}^{k_{1}}-t_{1}^{k_{1}})^{\left(\mu _{1}-1\right)/k_{1}}\dots (\lambda ^{k_{n}}x_{n}^{k_{n}}-t_{n}^{k_{n}})^{\left(\mu _{n}-1\right)/k_{n}}f(t_{1},...,t_{n})\,dt_{1}\dots dt_{n}=$ ${\ displaystyle \ int _ {0} ^ {\ lambda x_ {1}} \ dots \ int _ {0} ^ {\ lambda x_ {n}} (\ lambda ^ {k_ {1}} x_ {1} ^ {k_ {1}} - t_ {1} ^ {k_ {1}}) ^ {\ left (\ mu _ {1} -1 \ right) / k_ {1}} \ dots (\ lambda ^ {k_ { n}} x_ {n} ^ {k_ {n}} - t_ {n} ^ {k_ {n}}) ^ {\ left (\ mu _ {n} -1 \ right) / k_ {n}} f (t_ {1}, ..., t_ {n}) \, dt_ {1} \ dots dt_ {n} =}$ $\lambda ^{n}\int _{0}^{x_{1}}\dots \int _{0}^{x_{n}}(\lambda ^{k_{1}}x_{1}^{k_{1}}-\lambda ^{k_{1}}t_{1}'^{k_{1}})^{\left(\mu _{1}-1\right)/k_{1}}\dots (\lambda ^{k_{n}}x_{n}^{k_{n}}-\lambda ^{k_{n}}t_{n}'^{k_{n}})^{\left(\mu _{n}-1\right)/k_{n}}f(\lambda t_{1}',...,\lambda t_{n}')\,dt_{1}'\dots dt_{n}'=$ ${\ displaystyle \ lambda ^ {n} \ int _ {0} ^ {x_ {1}} \ dots \ int _ {0} ^ {x_ {n}} (\ lambda ^ {k_ {1}} x_ {1 } ^ {k_ {1}} - \ lambda ^ {k_ {1}} t_ {1} '^ {k_ {1}}) ^ {\ left (\ mu _ {1} -1 \ right) / k_ { 1}} \ dots (\ lambda ^ {k_ {n}} x_ {n} ^ {k_ {n}} - \ lambda ^ {k_ {n}} t_ {n} '^ {k_ {n}}) ^ {\ left (\ mu _ {n} -1 \ right) / k_ {n}} f (\ lambda t_ {1} ', ..., \ lambda t_ {n}') \, dt_ {1} ' \ dots dt_ {n} '=}$ $\lambda ^{q+\mu _{1}+\dots +\mu _{n}}\int _{0}^{x_{1}}\dots \int _{0}^{x_{n}}(x_{1}^{k_{1}}-t_{1}'^{k_{1}})^{\left(\mu _{1}-1\right)/k_{1}}\dots (x_{n}^{k_{n}}-t_{n}'^{k_{n}})^{\left(\mu _{n}-1\right)/k_{n}}f(t_{1}',...,t_{n}')\,dt_{1}'\dots dt_{n}'=$ ${\ displaystyle \ lambda ^ {q + \ mu _ {1} + \ dots + \ mu _ {n}} \ int _ {0} ^ {x_ {1}} \ dots \ int _ {0} ^ {x_ { n}} (x_ {1} ^ {k_ {1}} - t_ {1} '^ {k_ {1}}) ^ {\ left (\ mu _ {1} -1 \ right) / k_ {1} } \ dots (x_ {n} ^ {k_ {n}} - t_ {n} '^ {k_ {n}}) ^ {\ left (\ mu _ {n} -1 \ right) / k_ {n} } f (t_ {1} ', ..., t_ {n}') \, dt_ {1} '\ dots dt_ {n}' =}$ $\lambda ^{q+\mu _{1}+\dots +\mu _{n}}H(x_{1},x_{2},...,x_{n})$ ${\ displaystyle \ lambda ^ {q + \ mu _ {1} + \ dots + \ mu _ {n}} H (x_ {1}, x_ {2}, ..., x_ {n})}$ where the change of integration variables is made $t_{k}=\lambda t_{k}'$ ${\ displaystyle t_ {k} = \ lambda t_ {k} '}$ . (Note: convolution is possible only in part of variables.)

Theorem Any homogeneous function with a uniformity order $q$ ${\ displaystyle q}$ may be presented in the form

f(x_{1},x_{2},...,x_{n})=x_{1}^{q}\cdot h(x_{2}/x_{1},x_{3}/x_{1},...,x_{n}/x_{1}),

{\ displaystyle f (x_ {1}, x_ {2}, ..., x_ {n}) = x_ {1} ^ {q} \ cdot h (x_ {2} / x_ {1}, x_ {3 } / x_ {1}, ..., x_ {n} / x_ {1}),}

Where $h(t_{2},t_{3},...,t_{n})$ ${\ displaystyle h (t_ {2}, t_ {3}, ..., t_ {n})}$ - some function $(n-1)$ ${\ displaystyle (n-1)}$ variables. Any absolutely homogeneous function with a uniformity order $q$ ${\ displaystyle q}$ can be represented as

f(x_{1},x_{2},...,x_{n})=|x|^{q}\cdot h(x_{2}/x_{1},x_{3}/x_{1},...,x_{n}/x_{1}),

{\ displaystyle f (x_ {1}, x_ {2}, ..., x_ {n}) = | x | ^ {q} \ cdot h (x_ {2} / x_ {1}, x_ {3} /x_{1►,...,x_{n►/x_{1}),}

Where $h(t_{2},t_{3},...,t_{n})$ ${\ displaystyle h (t_ {2}, t_ {3}, ..., t_ {n})}$ - some function $(n-1)$ ${\ displaystyle (n-1)}$ variables.

Evidence.

Take a homogeneous function $g(x_{1},x_{2},...,x_{n})$ ${\ displaystyle g (x_ {1}, x_ {2}, ..., x_ {n})}$ zero degree. Then when choosing $\lambda =1/x_{1}$ ${\ displaystyle \ lambda = 1 / x_ {1}}$ we get a particular version of the desired ratio:

g(x_{1},x_{2},...,x_{n})=g(\lambda x_{1},\lambda x_{2},...,\lambda x_{n})=g(1,x_{2}/x_{1},...,x_{n}/x_{1})=h(x_{2}/x_{1},...,x_{n}/x_{1}).

{\ displaystyle g (x_ {1}, x_ {2}, ..., x_ {n}) = g (\ lambda x_ {1}, \ lambda x_ {2}, ..., \ lambda x_ {n }) = g (1, x_ {2} / x_ {1}, ..., x_ {n} / x_ {1}) = h (x_ {2} / x_ {1}, ..., x_ { n} / x_ {1}).}

For homogeneous function $f(x_{1},x_{2},...,x_{n})$ ${\ displaystyle f (x_ {1}, x_ {2}, ..., x_ {n})}$ degrees of $q\neq 0,$ ${\ displaystyle q \ neq 0,}$ function $g(x_{1},x_{2},...,x_{n})=f(x_{1},x_{2},...,x_{n})/x_{1}^{q}$ ${\ displaystyle g (x_ {1}, x_ {2}, ..., x_ {n}) = f (x_ {1}, x_ {2}, ..., x_ {n}) / x_ {1 } ^ {q}}$ will be a homogeneous function of degree zero. therefore $g(x_{1},...,x_{n})=h(x_{2}/x_{1},...,x_{n}/x_{1})$ ${\ displaystyle g (x_ {1}, ..., x_ {n}) = h (x_ {2} / x_ {1}, ..., x_ {n} / x_ {1})}$ and $f(x_{1},...,x_{n})=x_{1}^{q}\cdot h(x_{2}/x_{1},...,x_{n}/x_{1}).$ ${\ displaystyle f (x_ {1}, ..., x_ {n}) = x_ {1} ^ {q} \ cdot h (x_ {2} / x_ {1}, ..., x_ {n} / x_ {1}).}$

Consequence Any homogeneous degree function $q$ ${\ displaystyle q}$ (absolutely homogeneous degree function $q$ ${\ displaystyle q}$ ) can be represented in the form

f(x_{1},x_{2},...,x_{n})=\phi (x_{1},x_{2},...,x_{n})\cdot h(\phi _{1}(x_{1},x_{2},...,x_{n}),\phi _{2}(x_{1},x_{2},...,x_{n}),...,\phi _{n-1}(x_{1},x_{2},...,x_{n})),

{\ displaystyle f (x_ {1}, x_ {2}, ..., x_ {n}) = \ phi (x_ {1}, x_ {2}, ..., x_ {n}) \ cdot h (\ phi _ {1} (x_ {1}, x_ {2}, ..., x_ {n}), \ phi _ {2} (x_ {1}, x_ {2}, ..., x_ {n}), ..., \ phi _ {n-1} (x_ {1}, x_ {2}, ..., x_ {n})),}

Where $h(t_{1},t_{2},...,t_{n-1})$ ${\ displaystyle h (t_ {1}, t_ {2}, ..., t_ {n-1})}$ - some suitable function $(n-1)$ ${\ displaystyle (n-1)}$ variables $\phi (x_{1},x_{2},...,x_{n})$ ${\ displaystyle \ phi (x_ {1}, x_ {2}, ..., x_ {n})}$ - fixed homogeneous degree function $q$ ${\ displaystyle q}$ (fixed absolutely homogeneous degree function $q$ ${\ displaystyle q}$ ), but $\phi _{1}(x_{1},x_{2},...,x_{n}),$ ${\ displaystyle \ phi _ {1} (x_ {1}, x_ {2}, ..., x_ {n}),}$ $\phi _{2}(x_{1},x_{2},...,x_{n}))$ ${\ displaystyle \ phi _ {2} (x_ {1}, x_ {2}, ..., x_ {n}))}$ , ..., $\phi _{n-1}(x_{1},x_{2},...,x_{n}))$ ${\ displaystyle \ phi _ {n-1} (x_ {1}, x_ {2}, ..., x_ {n}))}$ - fixed functionally independent homogeneous functions of degree zero. With a fixed selection of functions $\phi ,\phi _{1},\phi _{2},...,\phi _{n-1}$ ${\ displaystyle \ phi, \ phi _ {1}, \ phi _ {2}, ..., \ phi _ {n-1}}$ this representation defines a one-to-one correspondence between homogeneous functions $f(x_{1},x_{2},...,x_{n})$ ${\ displaystyle f (x_ {1}, x_ {2}, ..., x_ {n})}$ degrees of $q$ ${\ displaystyle q}$ from $n$ ${\ displaystyle n}$ variables and functions $h(t_{1},t_{2},...,t_{n-1})$ ${\ displaystyle h (t_ {1}, t_ {2}, ..., t_ {n-1})}$ from $(n-1)$ ${\ displaystyle (n-1)}$ variables.

Euler 's theorem for homogeneous functions . To differentiable function $f(x_{1},x_{2},...,x_{n})$ ${\ displaystyle f (x_ {1}, x_ {2}, ..., x_ {n})}$ was a homogeneous function with a uniformity order $q,$ ${\ displaystyle q,}$ the fulfillment of the Euler relation is necessary and sufficient

\sum x_{k}f'_{x_{k}}(x_{1},x_{2},...,x_{n})=qf(x_{1},x_{2},...,x_{n}).

{\ displaystyle \ sum x_ {k} f '_ {x_ {k}} (x_ {1}, x_ {2}, ..., x_ {n}) = qf (x_ {1}, x_ {2} , ..., x_ {n}).}

Evidence.

Necessity is obtained from differentiation of equality $(*)$ ${\ displaystyle (*)}$ at $\lambda =1.$ ${\ displaystyle \ lambda = 1.}$ To prove sufficiency, we take the function $\varphi (\lambda )=\lambda ^{-q}f(\lambda x_{1},\lambda x_{2},...,\lambda x_{n})$ ${\ displaystyle \ varphi (\ lambda) = \ lambda ^ {- q} f (\ lambda x_ {1}, \ lambda x_ {2}, ..., \ lambda x_ {n})}$ with "frozen" $x_{1},x_{2},...,x_{n}.$ ${\ displaystyle x_ {1}, x_ {2}, ..., x_ {n}.}$ Differentiate it by $\lambda :$ ${\ displaystyle \ lambda:}$

\varphi '(\lambda )=-q\lambda ^{-q-1}f(\lambda x_{1},\lambda x_{2},...,\lambda x_{n})+\lambda ^{-q}\sum f'_{x_{k}}(\lambda x_{1},\lambda x_{2},...,\lambda x_{n})x_{k}.

{\ displaystyle \ varphi '(\ lambda) = - q \ lambda ^ {- q-1} f (\ lambda x_ {1}, \ lambda x_ {2}, ..., \ lambda x_ {n}) + \ lambda ^ {- q} \ sum f '_ {x_ {k}} (\ lambda x_ {1}, \ lambda x_ {2}, ..., \ lambda x_ {n}) x_ {k}.}

By virtue of the condition $\sum (\lambda x_{k})\cdot f'_{x_{k}}(\lambda x_{1},\lambda x_{2},...,\lambda x_{n})=qf(\lambda x_{1},\lambda x_{2},...,\lambda x_{n})$ ${\ displaystyle \ sum (\ lambda x_ {k}) \ cdot f '_ {x_ {k}} (\ lambda x_ {1}, \ lambda x_ {2}, ..., \ lambda x_ {n}) = qf (\ lambda x_ {1}, \ lambda x_ {2}, ..., \ lambda x_ {n})}$ we get $\varphi '(\lambda )=0$ ${\ displaystyle \ varphi '(\ lambda) = 0}$ and $\varphi (\lambda )=c=const.$ ${\ displaystyle \ varphi (\ lambda) = c = const.}$ Constant $c$ ${\ displaystyle c}$ determined from the condition $\varphi (1)=f(x_{1},x_{2},...,x_{n}).$ ${\ displaystyle \ varphi (1) = f (x_ {1}, x_ {2}, ..., x_ {n}).}$ As a result $\lambda ^{q}\varphi (\lambda )=f(\lambda x_{1},\lambda x_{2},...,\lambda x_{n})=\lambda ^{q}f(x_{1},x_{2},...,x_{n}).$ ${\ displaystyle \ lambda ^ {q} \ varphi (\ lambda) = f (\ lambda x_ {1}, \ lambda x_ {2}, ..., \ lambda x_ {n}) = \ lambda ^ {q} f (x_ {1}, x_ {2}, ..., x_ {n}).}$

Consequence If the function is differentiable and at each point in space the homogeneity relation $(*)$ ${\ displaystyle (*)}$ valid in a certain range of values $\lambda \in \left[\lambda _{0}-\varepsilon ,\lambda _{0}+\varepsilon \right]\subset \left[0,\infty \right),$ ${\ displaystyle \ lambda \ in \ left [\ lambda _ {0} - \ varepsilon, \ lambda _ {0} + \ varepsilon \ right] \ subset \ left [0, \ infty \ right),}$ then it is fair to all $\lambda >0.$ ${\ displaystyle \ lambda> 0.}$

Evidence.

We differentiate the relation $(*)$ ${\ displaystyle (*)}$ by $\lambda$ ${\ displaystyle \ lambda}$ at the point $\lambda =\lambda _{0}:$ ${\ displaystyle \ lambda = \ lambda _ {0}:}$

\sum x_{k}f'_{x_{k}}(\lambda _{0}x_{1},\lambda _{0}x_{2},...,\lambda _{0}x_{n})=q\lambda _{0}^{q-1}f(x_{1},x_{2},...,x_{n})={\frac {q}{\lambda _{0}}}f(\lambda _{0}x_{1},\lambda _{0}x_{2},...,\lambda _{0}x_{n}).

{\ displaystyle \ sum x_ {k} f '_ {x_ {k}} (\ lambda _ {0} x_ {1}, \ lambda _ {0} x_ {2}, ..., \ lambda _ {0 } x_ {n}) = q \ lambda _ {0} ^ {q-1} f (x_ {1}, x_ {2}, ..., x_ {n}) = {\ frac {q} {\ lambda _ {0}}} f (\ lambda _ {0} x_ {1}, \ lambda _ {0} x_ {2}, ..., \ lambda _ {0} x_ {n}).}

This means that at the point $y_{k}=\lambda _{0}x_{k}$ ${\ displaystyle y_ {k} = \ lambda _ {0} x_ {k}}$ the Euler relation holds, and due to the arbitrariness of the point $(x_{1},x_{2},...,x_{n})$ ${\ displaystyle (x_ {1}, x_ {2}, ..., x_ {n})}$ point $(y_{1},y_{2},...,y_{n})$ ${\ displaystyle (y_ {1}, y_ {2}, ..., y_ {n})}$ also arbitrary. Repeating the above proof of the Euler homogeneous function theorem, we get that at the point $(y_{1},y_{2},...,y_{n})$ ${\ displaystyle (y_ {1}, y_ {2}, ..., y_ {n})}$ the homogeneity relation holds, and for an arbitrary $\lambda >0.$ ${\ displaystyle \ lambda> 0.}$ Point $(x_{1},x_{2},...,x_{n})$ ${\ displaystyle (x_ {1}, x_ {2}, ..., x_ {n})}$ can choose so that the point $(y_{1},y_{2},...,y_{n})$ ${\ displaystyle (y_ {1}, y_ {2}, ..., y_ {n})}$ coincides with any given point in space. Therefore, at each point in space, the relation $(*)$ ${\ displaystyle (*)}$ performed at any $\lambda >0.$ ${\ displaystyle \ lambda> 0.}$

Lambda-homogeneous functions

Let a vector be given $\mathbf {\lambda } =(\lambda _{1},\lambda _{2},...,\lambda _{n}).$ ${\ displaystyle \ mathbf {\ lambda} = (\ lambda _ {1}, \ lambda _ {2}, ..., \ lambda _ {n}.}$ Function $n$ ${\ displaystyle n}$ variables $f(x_{1},x_{2},...,x_{n})$ ${\ displaystyle f (x_ {1}, x_ {2}, ..., x_ {n})}$ called $\lambda$ ${\ displaystyle \ lambda}$ homogeneous with homogeneity order $q$ ${\ displaystyle q}$ if for any $t>0$ ${\ displaystyle t> 0}$ and any $\mathbf {x} =(x_{1},x_{2},...,x_{n})\in {\mathbb {R} }^{n}$ ${\ displaystyle \ mathbf {x} = (x_ {1}, x_ {2}, ..., x_ {n}) \ in {\ mathbb {R}} ^ {n}}$ fair identity

f(t^{\lambda _{1}}x_{1},t^{\lambda _{2}}x_{2},...,t^{\lambda _{n}}x_{n})=t^{q}f(x_{1},x_{2},...,x_{n}).

{\ displaystyle f (t ^ {\ lambda _ {1}} x_ {1}, t ^ {\ lambda _ {2}} x_ {2}, ..., t ^ {\ lambda _ {n}} x_ {n}) = t ^ {q} f (x_ {1}, x_ {2}, ..., x_ {n}).}

At $\lambda _{k}=1$ ${\ displaystyle \ lambda _ {k} = 1}$ $\lambda$ ${\ displaystyle \ lambda}$ -homogeneous functions pass into ordinary homogeneous functions. Sometimes instead of uniformity $q$ ${\ displaystyle q}$ introduce a degree of uniformity $m$ ${\ displaystyle m}$ determined from the relation

f(t^{\lambda _{1}}x_{1},t^{\lambda _{2}}x_{2},...,t^{\lambda _{n}}x_{n})=t^{m{\frac {|\mathbf {\lambda } |}{n}}}f(x_{1},x_{2},...,x_{n}),

{\ displaystyle f (t ^ {\ lambda _ {1}} x_ {1}, t ^ {\ lambda _ {2}} x_ {2}, ..., t ^ {\ lambda _ {n}} x_ {n}) = t ^ {m {\ frac {| \ mathbf {\ lambda} |} {n}}} f (x_ {1}, x_ {2}, ..., x_ {n}),}

Where $|\mathbf {\lambda } |=\sum |\lambda _{k}|.$ ${\ displaystyle | \ mathbf {\ lambda} | = \ sum | \ lambda _ {k} |.}$ For ordinary homogeneous functions, the order of homogeneity $q$ ${\ displaystyle q}$ and degree of homogeneity $m$ ${\ displaystyle m}$ match.

If the private derivatives $f'_{x_{k}}(x_{1},x_{2},...,x_{n})$ ${\ displaystyle f '_ {x_ {k}} (x_ {1}, x_ {2}, ..., x_ {n})}$ continuous in $\mathbb {R} ^{n}$ ${\ displaystyle \ mathbb {R} ^ {n}}$ then for $\lambda$ ${\ displaystyle \ lambda}$ -uniform functions, the relation generalizing the Euler relation and obtained by differentiating the identities for $\lambda$ ${\ displaystyle \ lambda}$ -uniformities at a point $t=1$ ${\ displaystyle t = 1}$ :

\sum \lambda _{x}x_{k}f'_{x_{k}}(x_{1},x_{2},...,x_{n})=qf(x_{1},x_{2},...,x_{n}).

{\ displaystyle \ sum \ lambda _ {x} x_ {k} f '_ {x_ {k}} (x_ {1}, x_ {2}, ..., x_ {n}) = qf (x_ {1 }, x_ {2}, ..., x_ {n}).}

As in the case of ordinary homogeneous functions, this relation is necessary and sufficient for the function $f(x_{1},x_{2},...,x_{n})$ ${\ displaystyle f (x_ {1}, x_ {2}, ..., x_ {n})}$ was $\lambda$ ${\ displaystyle \ lambda}$ -uniform function with vector $(\lambda _{1},\lambda _{2},...,\lambda _{n})$ ${\ displaystyle (\ lambda _ {1}, \ lambda _ {2}, ..., \ lambda _ {n})}$ and uniformity $q .$ ${\ displaystyle q.}$ To prove sufficiency, we must consider the function $\varphi (t)=t^{-q}f(t^{\lambda _{1}}x_{1},t^{\lambda _{2}}x_{2},...,t^{\lambda _{n}}x_{n})$ ${\ displaystyle \ varphi (t) = t ^ {- q} f (t ^ {\ lambda _ {1}} x_ {1}, t ^ {\ lambda _ {2}} x_ {2}, ... , t ^ {\ lambda _ {n}} x_ {n})}$ and make sure that when the indicated differential relation is satisfied, its derivative is equal to zero, that is, that this function is constant and that $\varphi (t)\equiv \varphi (1).$ ${\ displaystyle \ varphi (t) \ equiv \ varphi (1).}$

If $f(x_{1},x_{2},...,x_{n})$ ${\ displaystyle f (x_ {1}, x_ {2}, ..., x_ {n})}$ - $\lambda$ ${\ displaystyle \ lambda}$ -uniform function with a vector $\mathbf {\lambda } =(\lambda _{1},\lambda _{2},...,\lambda _{n})$ ${\ displaystyle \ mathbf {\ lambda} = (\ lambda _ {1}, \ lambda _ {2}, ..., \ lambda _ {n})}$ and uniformity $q$ ${\ displaystyle q}$ then she is $\lambda$ ${\ displaystyle \ lambda}$ -uniform function with vector $\mathbf {\lambda } =(\alpha \lambda _{1},\alpha \lambda _{2},...,\alpha \lambda _{n})$ ${\ displaystyle \ mathbf {\ lambda} = (\ alpha \ lambda _ {1}, \ alpha \ lambda _ {2}, ..., \ alpha \ lambda _ {n})}$ and uniformity $\alpha q$ ${\ displaystyle \ alpha q}$ (follows from substituting into the identity for $\lambda$ ${\ displaystyle \ lambda}$ -homogeneities of the new parameter $t'\to t^{\alpha }$ ${\ displaystyle t '\ to t ^ {\ alpha}}$ ) Therefore, when considering $\lambda$ ${\ displaystyle \ lambda}$ -homogeneous functions is enough to be limited to $\sum |\lambda _{k}|=const.$ ${\ displaystyle \ sum | \ lambda _ {k} | = const.}$ In particular, normalization $\sum |\lambda _{k}|$ ${\ displaystyle \ sum | \ lambda _ {k} |}$ can be chosen so that the order of homogeneity $q$ ${\ displaystyle q}$ was equal to a pre-fixed value. In addition, without loss of generality, we can assume that $\lambda _{k}\neq 0.$ ${\ displaystyle \ lambda _ {k} \ neq 0.}$

When replacing variables $x_{k}=y_{k}^{\lambda _{k}}$ ${\ displaystyle x_ {k} = y_ {k} ^ {\ lambda _ {k}}}$ $\lambda$ ${\ displaystyle \ lambda}$ -uniform function $f(x_{1},x_{2},...,x_{n})$ ${\ displaystyle f (x_ {1}, x_ {2}, ..., x_ {n})}$ with vector $\mathbf {\lambda } =(\lambda _{1},\lambda _{2},...,\lambda _{n})$ ${\ displaystyle \ mathbf {\ lambda} = (\ lambda _ {1}, \ lambda _ {2}, ..., \ lambda _ {n})}$ and homogeneity order $q$ ${\ displaystyle q}$ goes into a regular homogeneous function $g(y_{1},y_{2},...,y_{n})$ ${\ displaystyle g (y_ {1}, y_ {2}, ..., y_ {n})}$ with uniformity order $q$ ${\ displaystyle q}$ . It follows that the general idea for $\lambda$ ${\ displaystyle \ lambda}$ -uniform functions with a vector $\mathbf {\lambda } =(\lambda _{1},\lambda _{2},...,\lambda _{n})$ ${\ displaystyle \ mathbf {\ lambda} = (\ lambda _ {1}, \ lambda _ {2}, ..., \ lambda _ {n})}$ and uniformity $q$ ${\ displaystyle q}$ has the form:

f(x_{1},x_{2},...,x_{n})=x_{1}^{q/\lambda _{1}}\cdot h(x_{2}^{1/\lambda _{2}}/x_{1}^{1/\lambda _{1}},x_{3}^{1/\lambda _{3}}/x_{1}^{1/\lambda _{1}},\ldots ,x_{n}^{1/\lambda _{n}}/x_{1}^{1/\lambda _{1}}),

{\ displaystyle f (x_ {1}, x_ {2}, ..., x_ {n}) = x_ {1} ^ {q / \ lambda _ {1}} \ cdot h (x_ {2} ^ { 1 / \ lambda _ {2}} / x_ {1} ^ {1 / \ lambda _ {1}}, x_ {3} ^ {1 / \ lambda _ {3}} / x_ {1} ^ {1 / \ lambda _ {1}}, \ ldots, x_ {n} ^ {1 / \ lambda _ {n}} / x_ {1} ^ {1 / \ lambda _ {1}}),}

Where $h(t_{2},t_{3},...,t_{n})$ ${\ displaystyle h (t_ {2}, t_ {3}, ..., t_ {n})}$ - some function $(n-1)$ ${\ displaystyle (n-1)}$ variables.

Source: Ya. S. Bugrov, S. M. Nikolsky, Higher mathematics: textbook for universities (3 volumes), Volume 2: Differential and integral calculus ( http://www.sernam.ru/lect_math2.php ) , section 8.8.4.

Euler Operator

Differential operator

x_{1}{\frac {\partial f}{\partial x_{1}}}+x_{2}{\frac {\partial f}{\partial x_{2}}}+\ldots +x_{n}{\frac {\partial f}{\partial x_{n}}}

{\ displaystyle x_ {1} {\ frac {\ partial f} {\ partial x_ {1}}} + x_ {2} {\ frac {\ partial f} {\ partial x_ {2}}} + \ ldots + x_ {n} {\ frac {\ partial f} {\ partial x_ {n}}}}

sometimes called the Euler operator, by analogy with the Euler identity for homogeneous functions. From the Euler theorem for homogeneous functions given above, it follows that the eigenfunctions of this operator are homogeneous functions and only they, and the eigenvalue for such a function is its homogeneity order.

Accordingly, the functions that turn the Euler operator into a constant are the logarithms of homogeneous functions and only they. The functions that turn the Euler operator to zero are homogeneous functions of zero order and only they (the logarithm of a homogeneous function of zero order is itself a homogeneous function of zero order).

Similarly for a differential operator

\lambda _{1}x_{1}{\frac {\partial f}{\partial x_{1}}}+\lambda _{2}x_{2}{\frac {\partial f}{\partial x_{2}}}+\ldots +\lambda _{n}x_{n}{\frac {\partial f}{\partial x_{n}}}

{\ displaystyle \ lambda _ {1} x_ {1} {\ frac {\ partial f} {\ partial x_ {1}}} + \ lambda _ {2} x_ {2} {\ frac {\ partial f} { \ partial x_ {2}}} + \ ldots + \ lambda _ {n} x_ {n} {\ frac {\ partial f} {\ partial x_ {n}}}}

own functions are $\lambda$ ${\ displaystyle \ lambda}$ -uniform functions with a vector $(\lambda _{1},\lambda _{2},\ldots ,\lambda _{n})$ ${\ displaystyle (\ lambda _ {1}, \ lambda _ {2}, \ ldots, \ lambda _ {n})}$ and only they, and the eigenvalue is the homogeneity order $\lambda$ ${\ displaystyle \ lambda}$ -uniform function. The logarithms turn this constant into a constant $\lambda$ ${\ displaystyle \ lambda}$ -uniform functions with a vector $(\lambda _{1},\lambda _{2},\ldots ,\lambda _{n})$ ${\ displaystyle (\ lambda _ {1}, \ lambda _ {2}, \ ldots, \ lambda _ {n})}$ , and no other features.

A further generalization of the Euler operator is the differential operator

\lambda _{1}x_{1}^{\mu _{1}}{\frac {\partial f}{\partial x_{1}}}+\lambda _{2}x_{2}^{\mu _{2}}{\frac {\partial f}{\partial x_{2}}}+\ldots +\lambda _{n}x_{n}^{\mu _{n}}{\frac {\partial f}{\partial x_{n}}},

{\ displaystyle \ lambda _ {1} x_ {1} ^ {\ mu _ {1}} {\ frac {\ partial f} {\ partial x_ {1}}} + \ lambda _ {2} x_ {2} ^ {\ mu _ {2}} {\ frac {\ partial f} {\ partial x_ {2}}} + \ ldots + \ lambda _ {n} x_ {n} ^ {\ mu _ {n}} { \ frac {\ partial f} {\ partial x_ {n}}},}

which reduces to the Euler operator $y_{1}{\frac {\partial f}{\partial y_{1}}}+y_{2}{\frac {\partial f}{\partial y_{2}}}+\ldots +y_{n}{\frac {\partial f}{\partial y_{n}}}$ ${\ displaystyle y_ {1} {\ frac {\ partial f} {\ partial y_ {1}}} + y_ {2} {\ frac {\ partial f} {\ partial y_ {2}}} + \ ldots + y_ {n} {\ frac {\ partial f} {\ partial y_ {n}}}}$ replacement $y_{k}=\exp \left({\frac {x^{1-\mu _{k}}}{\lambda _{k}\left(1-\mu _{k}\right)}}\right)$ ${\ displaystyle y_ {k} = \ exp \ left ({\ frac {x ^ {1- \ mu _ {k}}} {\ lambda _ {k} \ left (1- \ mu _ {k} \ right )}} \ right)}$ at $\mu _{k}\neq 1;$ ${\ displaystyle \ mu _ {k} \ neq 1;}$ $y_{k}=x^{1/\lambda _{k}}$ ${\ displaystyle y_ {k} = x ^ {1 / \ lambda _ {k}}}$ at $\mu _{k}=1.$ ${\ displaystyle \ mu _ {k} = 1.}$ Also to the Euler operator using replacement $y_{k}=\exp \left(\int _{a_{k}}^{x}{\frac {dt}{h_{k}(t)}}\right)$ ${\ displaystyle y_ {k} = \ exp \ left (\ int _ {a_ {k}} ^ {x} {\ frac {dt} {h_ {k} (t)}} \ right)}$ all differential operators of the form $h_{1}(x_{1}){\frac {\partial f}{\partial x_{1}}}+h_{2}(x_{2}){\frac {\partial f}{\partial x_{2}}}+\ldots +h_{n}(x_{n}){\frac {\partial f}{\partial x_{n}}}.$ ${\ displaystyle h_ {1} (x_ {1}) {\ frac {\ partial f} {\ partial x_ {1}}} + h_ {2} (x_ {2}) {\ frac {\ partial f} { \ partial x_ {2}}} + \ ldots + h_ {n} (x_ {n}) {\ frac {\ partial f} {\ partial x_ {n}}}.}$

Source: Chi Woo, Igor Khavkine, Euler's theorem on homogeneous functions ( PlanetMath.org )

Limited homogeneous functions

Function $f(x_{1},x_{2},\ldots ,x_{n}):\mathbb {R} ^{n}\to \mathbb {R}$ ${\ displaystyle f (x_ {1}, x_ {2}, \ ldots, x_ {n}): \ mathbb {R} ^ {n} \ to \ mathbb {R}}$ called limited homogeneity with an indicator of homogeneity $q$ ${\ displaystyle q}$ with respect to the set of positive real numbers $\Lambda$ ${\ displaystyle \ Lambda}$ (called the homogeneity set) if for all ${\vec {x}}\in \mathbb {R} ^{n}$ ${\ displaystyle {\ vec {x}} \ in \ mathbb {R} ^ {n}}$ and for everyone $\lambda \in \Lambda$ ${\ displaystyle \ lambda \ in \ Lambda}$ fair identity

f(\lambda {\vec {x}})=\lambda ^{q}f({\vec {x}}).

{\ displaystyle f (\ lambda {\ vec {x}}) = \ lambda ^ {q} f ({\ vec {x}}).}

Lots of uniformity $\Lambda$ ${\ displaystyle \ Lambda}$ always contains a unit. Lots of uniformity $\Lambda$ ${\ displaystyle \ Lambda}$ cannot include an arbitrarily small continuous segment $\lambda \in \left[\lambda _{0}-\varepsilon ,\lambda _{0}+\varepsilon \right]$ ${\ displaystyle \ lambda \ in \ left [\ lambda _ {0} - \ varepsilon, \ lambda _ {0} + \ varepsilon \ right]}$ - otherwise, a boundedly homogeneous function turns out to be an ordinary homogeneous function (see below the section “Some functional equations associated with homogeneous functions”). Therefore, those boundedly homogeneous functions for which $\Lambda \neq \{1\}$ ${\ displaystyle \ Lambda \ neq \ {1 \}}$ and which have a lot of uniformity $\Lambda$ ${\ displaystyle \ Lambda}$ purely discrete.

Example 1. Function $f(x)=x^{q}\sin(\log |x|)$ ${\ displaystyle f (x) = x ^ {q} \ sin (\ log | x |)}$ is uniformly homogeneous with an indicator of uniformity $q$ ${\ displaystyle q}$ relatively many $\Lambda =\{e^{2\pi m}\},$ ${\ displaystyle \ Lambda = \ {e ^ {2 \ pi m} \},}$ Where $m$ ${\ displaystyle m}$ - whole numbers.

Example 2. Function $f(x,y,z)=(x^{2}+2y^{2}+3z^{2})^{q/2}\cos(\log {\sqrt {x^{2}-xy+y^{2}}})$ ${\ displaystyle f (x, y, z) = (x ^ {2} + 2y ^ {2} + 3z ^ {2}) ^ {q / 2} \ cos (\ log {\ sqrt {x ^ {2 } -xy + y ^ {2}}})}$ is uniformly homogeneous with an indicator of uniformity $q$ ${\ displaystyle q}$ relatively many $\Lambda =\{e^{2\pi k}\},$ ${\ displaystyle \ Lambda = \ {e ^ {2 \ pi k} \},}$ Where $k$ ${\ displaystyle k}$ - whole numbers.

Theorem. To function $f(x_{1},x_{2},...,x_{n}),$ ${\ displaystyle f (x_ {1}, x_ {2}, ..., x_ {n}),}$ defined at $x_{1}>0,$ ${\ displaystyle x_ {1}> 0,}$ was boundedly homogeneous with a uniformity order $q,$ ${\ displaystyle q,}$ it is necessary and sufficient for it to look

f(x_{1},x_{2},...,x_{n})=x_{1}^{q}\cdot H(\log x_{1},x_{2}/x_{1},x_{3}/x_{1},\ldots ,x_{n}/x_{1}),

{\ displaystyle f (x_ {1}, x_ {2}, ..., x_ {n}) = x_ {1} ^ {q} \ cdot H (\ log x_ {1}, x_ {2} / x_ {1}, x_ {3} / x_ {1}, \ ldots, x_ {n} / x_ {1}),}

Where $H(y,t_{2},t_{3},\ldots ,t_{n})$ ${\ displaystyle H (y, t_ {2}, t_ {3}, \ ldots, t_ {n})}$ - function periodic in variable $y$ ${\ displaystyle y}$ with at least one period independent of $t_{2},t_{3},\ldots ,t_{n}.$ ${\ displaystyle t_ {2}, t_ {3}, \ ldots, t_ {n}.}$ In this case, a lot of uniformity $\Lambda$ ${\ displaystyle \ Lambda}$ consists of numbers $\{e^{Y_{k}}\},$ ${\ displaystyle \ {e ^ {Y_ {k}} \},}$ Where $Y_{k}$ ${\ displaystyle Y_ {k}}$ - function periods $H(y,t_{2},t_{3},\ldots ,t_{n}),$ ${\ displaystyle H (y, t_ {2}, t_ {3}, \ ldots, t_ {n}),}$ independent of $t_{2},t_{3},\ldots ,t_{n}.$ ${\ displaystyle t_ {2}, t_ {3}, \ ldots, t_ {n}.}$

Evidence. Sufficiency is checked directly, the need must be proved. Let's make a change of variables

x_{1},x_{2},...,x_{n}\to x_{1},t_{2},...,t_{n},

{\ displaystyle x_ {1}, x_ {2}, ..., x_ {n} \ to x_ {1}, t_ {2}, ..., t_ {n},}

Where

t_{k}=x_{k}/x_{1},

{\ displaystyle t_ {k} = x_ {k} / x_ {1},}

so that $f(x_{1},x_{2},...,x_{n})=g(x_{1},t_{2},...,t_{n}).$ ${\ displaystyle f (x_ {1}, x_ {2}, ..., x_ {n}) = g (x_ {1}, t_ {2}, ..., t_ {n}).}$ If we now consider the function $h(x_{1},t_{2},...,t_{n})=g(x_{1},t_{2},...,t_{n})/x_{1}^{q},$ ${\ displaystyle h (x_ {1}, t_ {2}, ..., t_ {n}) = g (x_ {1}, t_ {2}, ..., t_ {n}) / x_ {1 } ^ {q},}$ then from the homogeneity condition we obtain for all admissible $x_{1}$ ${\ displaystyle x_ {1}}$ equality

h(\lambda x_{1},t_{2},...,t_{n})=h(x_{1},t_{2},t_{3},...,t_{n}),

{\ displaystyle h (\ lambda x_ {1}, t_ {2}, ..., t_ {n}) = h (x_ {1}, t_ {2}, t_ {3}, ..., t_ { n}),}

which will be fair when $\lambda \in \Lambda .$ ${\ displaystyle \ lambda \ in \ Lambda.}$ If only a lot $\Lambda$ ${\ displaystyle \ Lambda}$ does not consist of only one unit, then after replacement $x_{1}=\exp(y)$ ${\ displaystyle x_ {1} = \ exp (y)}$ function

H(y,t_{2},...,t_{n})=H(\log x_{1},t_{2},...,t_{n})=h(x_{1},t_{2},...,t_{n})

{\ displaystyle H (y, t_ {2}, ..., t_ {n}) = H (\ log x_ {1}, t_ {2}, ..., t_ {n}) = h (x_ { 1}, t_ {2}, ..., t_ {n})}

turns out to be periodic in a variable $y$ ${\ displaystyle y}$ with non-zero period $\log \lambda$ ${\ displaystyle \ log \ lambda}$ for any selected in a fixed way $\lambda \in \Lambda ,$ ${\ displaystyle \ lambda \ in \ Lambda,}$ since from the above equality the relation

H(\log x_{1}+\log \lambda ,t_{2},...,t_{n})=H(\log x_{1},t_{2},...,t_{n}).

{\ displaystyle H (\ log x_ {1} + \ log \ lambda, t_ {2}, ..., t_ {n}) = H (\ log x_ {1}, t_ {2}, ..., t_ {n}).}

Obviously, the selected fixed value $\log \lambda$ ${\ displaystyle \ log \ lambda}$ will be a function period $H(y,t_{2},...,t_{n})$ ${\ displaystyle H (y, t_ {2}, ..., t_ {n})}$ at once for all $t_{2},...,t_{n}.$ ${\ displaystyle t_ {2}, ..., t_ {n}.}$

The consequences:

If there is the smallest positive period $Y>0,$ ${\ displaystyle Y> 0,}$ independent of $t_{2},t_{3},\ldots ,t_{n},$ ${\ displaystyle t_ {2}, t_ {3}, \ ldots, t_ {n},}$ then a lot of uniformity $\Lambda$ ${\ displaystyle \ Lambda}$ has the form $\{e^{mY}\},$ ${\ displaystyle \ {e ^ {mY} \},}$ Where $m=0,\pm 1,\pm 2,\dots$ ${\ displaystyle m = 0, \ pm 1, \ pm 2, \ dots}$ Are arbitrary integers. (If $Y$ ${\ displaystyle Y}$ - smallest positive period of the function $H(y,...),$ ${\ displaystyle H (y, ...),}$ that's all $Y_{m}=mY$ ${\ displaystyle Y_ {m} = mY}$ - its periods, therefore numbers $\{e^{mY}\}$ ${\ displaystyle \ {e ^ {mY} \}}$ will come in a lot of homogeneity. If there is such a value of homogeneity $\lambda _{*}=e^{Y_{*}},$ ${\ displaystyle \ lambda _ {*} = e ^ {Y _ {*}},}$ what $e^{mY}<e^{Y_{*}}<e^{(m+1)Y},$ ${\ displaystyle e ^ {mY} <e ^ {Y _ {*}} <e ^ {(m + 1) Y},}$ then $Y_{*}-mY$ ${\ displaystyle Y _ {*} - mY}$ will be a positive period independent of $t_{2},...,t_{n},$ ${\ displaystyle t_ {2}, ..., t_ {n},}$ which will be less than $Y .$ ${\ displaystyle Y.}$ )
If the function $H(y,\ldots )$ ${\ displaystyle H (y, \ ldots)}$ Is a constant on a variable $y,$ ${\ displaystyle y,}$ then she does not have the smallest positive period (any positive number is her period). In this case $H(y,\ldots )$ ${\ displaystyle H (y, \ ldots)}$ independent of variable $y,$ ${\ displaystyle y,}$ and function
$f(x_{1},x_{2},...,x_{n})=x_{1}^{q}\cdot H(x_{2}/x_{1},x_{3}/x_{1},\ldots ,x_{n}/x_{1})$ ${\ displaystyle f (x_ {1}, x_ {2}, ..., x_ {n}) = x_ {1} ^ {q} \ cdot H (x_ {2} / x_ {1}, x_ {3 } / x_ {1}, \ ldots, x_ {n} / x_ {1})}$
Is an ordinary positively homogeneous function (at least). Lots of uniformity $\Lambda$ ${\ displaystyle \ Lambda}$ in this case, the entire positive axis $\lambda >0$ ${\ displaystyle \ lambda> 0}$ (at least).
Exotic cases are possible when a periodic function $H(y,...)$ ${\ displaystyle H (y, ...)}$ there is no smallest positive period, but at the same time it is not a constant. For example, for a Dirichlet function equal to 1 at rational points and equal to 0 at irrational points, the period is any rational number. In this case, a lot of uniformity $\Lambda$ ${\ displaystyle \ Lambda}$ may have a fairly complex structure. However, if for each set of values $t_{2},t_{3},\ldots ,t_{n}$ ${\ displaystyle t_ {2}, t_ {3}, \ ldots, t_ {n}}$ have a periodic function $H(y,...)$ ${\ displaystyle H (y, ...)}$ there is a limit to the variable $y$ ${\ displaystyle y}$ at least at one point, this function either has the smallest positive period (and all other periods are multiples of the smallest positive period), or is a constant in a variable $y .$ ${\ displaystyle y.}$
Boundedly homogeneous functions defined for $x<0,$ ${\ displaystyle x <0,}$ have the form
$f(x_{1},x_{2},...,x_{n})=(-x_{1})^{q}\cdot H(\log(-x_{1}),x_{2}/x_{1},x_{3}/x_{1},\ldots ,x_{n}/x_{1})$ ${\ displaystyle f (x_ {1}, x_ {2}, ..., x_ {n}) = (- x_ {1}) ^ {q} \ cdot H (\ log (-x_ {1}), x_ {2} / x_ {1}, x_ {3} / x_ {1}, \ ldots, x_ {n} / x_ {1})}$
with properly selected function $H(y,t_{2},t_{3},\ldots ,t_{n}),$ ${\ displaystyle H (y, t_ {2}, t_ {3}, \ ldots, t_ {n}),}$ periodic in variable $y .$ ${\ displaystyle y.}$
Boundedly homogeneous functions defined on the entire numerical axis minus the point $x=0,$ ${\ displaystyle x = 0,}$ have the form
$f(x_{1},x_{2},...,x_{n})=|x_{1}|^{q}\cdot H_{\pm }(\log |x_{1}|,x_{2}/x_{1},x_{3}/x_{1},\ldots ,x_{n}/x_{1}),$ ${\ displaystyle f (x_ {1}, x_ {2}, ..., x_ {n}) = | x_ {1} | ^ {q} \ cdot H _ {\ pm} (\ log | x_ {1} |, x_ {2} / x_ {1}, x_ {3} / x_ {1}, \ ldots, x_ {n} / x_ {1}),}$
with properly selected function $H_{\pm }(y,t_{2},t_{3},\ldots ,t_{n}),$ ${\ displaystyle H _ {\ pm} (y, t_ {2}, t_ {3}, \ ldots, t_ {n}),}$ periodic in variable $y$ ${\ displaystyle y}$ (where the designation $H_{\pm }(\ldots )$ ${\ displaystyle H _ {\ pm} (\ ldots)}$ emphasizes that for the range of values $x_{1}>0$ ${\ displaystyle x_ {1}> 0}$ and for the range of values $x_{1}<0$ ${\ displaystyle x_ {1} <0}$ generally speaking, different periodic functions are chosen $H(y)$ ${\ displaystyle H (y)}$ each with a scope $y\in (-\infty ,+\infty )$ ${\ displaystyle y \ in (- \ infty, + \ infty)}$ , but always having at the same time the same period).
Formula $f(x_{1},...,x_{n})=x_{1}^{q}\cdot H(\log |x_{1}|,x_{2}/x_{1},\ldots ,x_{n}/x_{1}),$ ${\ displaystyle f (x_ {1}, ..., x_ {n}) = x_ {1} ^ {q} \ cdot H (\ log | x_ {1} |, x_ {2} / x_ {1} , \ ldots, x_ {n} / x_ {1}),}$ is universal, but does not reflect the equality of all variables. You can imagine a function $H(y,t_{2},\dots ,t_{n}\ldots )$ ${\ displaystyle H (y, t_ {2}, \ dots, t_ {n} \ ldots)}$ as $G\left(w\cdot y+\log W(t_{2},\dots ,t_{n}),t_{2},\dots ,t_{n}\right),$ ${\ displaystyle G \ left (w \ cdot y + \ log W (t_ {2}, \ dots, t_ {n}), t_ {2}, \ dots, t_ {n} \ right),}$ where is the period of the function $G\left(t,t_{2},\dots ,t_{n}\right)$ ${\ displaystyle G \ left (t, t_ {2}, \ dots, t_ {n} \ right)}$ is equal to $2\pi ,$ ${\ displaystyle 2 \ pi,}$ normalization factor $w$ ${\ displaystyle w}$ independent of $t_{2},\dots ,t_{n},$ ${\ displaystyle t_ {2}, \ dots, t_ {n},}$ and function $W(t_{2},\dots ,t_{n})$ ${\ displaystyle W (t_ {2}, \ dots, t_ {n})}$ selected fixed. With this notation, uniformly homogeneous functions take the form
$f(x_{1},...,x_{n})=F(\log Q(x_{1},\ldots ,x_{n}),x_{1},\ldots ,x_{n}),$ ${\ displaystyle f (x_ {1}, ..., x_ {n}) = F (\ log Q (x_ {1}, \ ldots, x_ {n}), x_ {1}, \ ldots, x_ { n}),}$
Where $F(y,x_{1},x_{2},\ldots ,x_{n})$ ${\ displaystyle F (y, x_ {1}, x_ {2}, \ ldots, x_ {n})}$ - homogeneous function with an indicator of homogeneity $q$ ${\ displaystyle q}$ by variables $x_{1},x_{2},\ldots ,x_{n}$ ${\ displaystyle x_ {1}, x_ {2}, \ ldots, x_ {n}}$ and periodic with a period $2\pi$ ${\ displaystyle 2 \ pi}$ by variable $y,$ ${\ displaystyle y,}$ $Q(x_{1},x_{2},\ldots ,x_{n}),$ ${\ displaystyle Q (x_ {1}, x_ {2}, \ ldots, x_ {n}),}$ - fixed homogeneous function with an indicator of homogeneity $w$ ${\ displaystyle w}$ by variables $x_{1},x_{2},\ldots ,x_{n},$ ${\ displaystyle x_ {1}, x_ {2}, \ ldots, x_ {n},}$ and the set of homogeneity has the form $\Lambda =\{e^{2\pi m/w}\},$ ${\ displaystyle \ Lambda = \ {e ^ {2 \ pi m / w} \},}$ Where $m=0,\pm 1,\pm 2,\dots$ ${\ displaystyle m = 0, \ pm 1, \ pm 2, \ dots}$ Are arbitrary integers.
Decomposing a Periodic Function $F(y,x_{1},\ldots ,x_{n})$ ${\ displaystyle F (y, x_ {1}, \ ldots, x_ {n})}$ from the previous paragraph in the Fourier series, we can get the expression
$A_{0}(x_{1},\ldots ,x_{n})+\sum A_{k}(x_{1},\ldots ,x_{n})\cos k\log Q(x_{1},\ldots ,x_{n})+B_{k}(x_{1},\ldots ,x_{n})\sin k\log Q(x_{1},\ldots ,x_{n}),$ ${\ displaystyle A_ {0} (x_ {1}, \ ldots, x_ {n}) + \ sum A_ {k} (x_ {1}, \ ldots, x_ {n}) \ cos k \ log Q (x_ {1}, \ ldots, x_ {n}) + B_ {k} (x_ {1}, \ ldots, x_ {n}) \ sin k \ log Q (x_ {1}, \ ldots, x_ {n} ),}$
Where $A_{k}(x_{1},\ldots ,x_{n})$ ${\ displaystyle A_ {k} (x_ {1}, \ ldots, x_ {n})}$ and $B_{k}(x_{1},\ldots ,x_{n})$ ${\ displaystyle B_ {k} (x_ {1}, \ ldots, x_ {n})}$ - arbitrary homogeneous functions with an indicator of homogeneity $q,$ ${\ displaystyle q,}$ $Q(x_{1},\ldots ,x_{n})$ ${\ displaystyle Q (x_ {1}, \ ldots, x_ {n})}$ - an arbitrary way fixed homogeneous function with an indicator of homogeneity $w,$ ${\ displaystyle w,}$ and a lot of uniformity $\Lambda =\{e^{mY}\},$ ${\ displaystyle \ Lambda = \ {e ^ {mY} \},}$ written as $\Lambda =\{e^{2\pi m/w}\},$ ${\ displaystyle \ Lambda = \ {e ^ {2 \ pi m / w} \},}$ Where $m$ ${\ displaystyle m}$ - whole numbers. This formula is the most general way of writing for piecewise continuous boundedly homogeneous functions with a uniformity order $q$ ${\ displaystyle q}$ and a lot of uniformity $\Lambda =\{e^{2\pi m/w}\}.$ ${\ displaystyle \ Lambda = \ {e ^ {2 \ pi m / w} \}.}$ In particular, replacing a fixed function $Q(x_{1},\ldots ,x_{n})$ ${\ displaystyle Q (x_ {1}, \ ldots, x_ {n})}$ to a set of arbitrary homogeneous functions $Q_{k}(x_{1},\ldots ,x_{n})$ ${\ displaystyle Q_ {k} (x_ {1}, \ ldots, x_ {n})}$ It will not add generality to this formula, but will only diversify the presentation form for the same boundedly homogeneous function.

Bibliography: Konrad Schlude, Bemerkung zu beschränkt homogenen Funktionen . - Elemente der Mathematik 54 (1999).

Source of information: J.Pahikkala. Boundedly homogeneous function ( PlanetMath.org ).

Attached homogeneous functions

[section not yet written]

Source: I. M. Gelfand, Z. Ya. Shapiro. Homogeneous functions and their applications. Advances in Mathematical Sciences, vol. 10 (1955) vol. 3, pp. 3–70.

Mutually homogeneous functions

[section not yet written]

Source: I. M. Gelfand, Z. Ya. Shapiro. Homogeneous functions and their applications. Advances in Mathematical Sciences, vol. 10 (1955) vol. 3, pp. 3–70.

Some functional equations related to homogeneous functions

1. Let

f\left(\lambda x_{1},\lambda x_{2},\dots ,\lambda x_{n}\right)=C\left(\lambda \right)f\left(x_{1},x_{2},\dots ,x_{n}\right)

{\ displaystyle f \ left (\ lambda x_ {1}, \ lambda x_ {2}, \ dots, \ lambda x_ {n} \ right) = C \ left (\ lambda \ right) f \ left (x_ {1 }, x_ {2}, \ dots, x_ {n} \ right)}

for some function $C\left(\lambda \right)$ ${\ displaystyle C \ left (\ lambda \ right)}$ on the interval $\lambda \in \left[\lambda _{0}-\varepsilon ,\lambda _{0}+\varepsilon \right].$ ${\ displaystyle \ lambda \ in \ left [\ lambda _ {0} - \ varepsilon, \ lambda _ {0} + \ varepsilon \ right].}$ What should be the function $f\left(x_{1},x_{2},\dots ,x_{n}\right)?$ ${\ displaystyle f \ left (x_ {1}, x_ {2}, \ dots, x_ {n} \ right)?}$

Decision. We differentiate both sides of this relation with respect to $\lambda .$ ${\ displaystyle \ lambda.}$ Get

x_{1}{\frac {\partial f(\lambda x_{1},\dots ,\lambda x_{n})}{\partial (\lambda x_{1})}}+x_{2}{\frac {\partial f(\lambda x_{1},\dots ,\lambda x_{n})}{\partial (\lambda x_{2})}}+\dots +x_{n}{\frac {\partial f(\lambda x_{1},\dots ,\lambda x_{n})}{\partial (\lambda x_{n})}}={\frac {\partial C\left(\lambda \right)}{\partial \lambda }}f(x_{1},\dots ,x_{n}).

{\ displaystyle x_ {1} {\ frac {\ partial f (\ lambda x_ {1}, \ dots, \ lambda x_ {n})} {\ partial (\ lambda x_ {1})}} + x_ {2 } {\ frac {\ partial f (\ lambda x_ {1}, \ dots, \ lambda x_ {n})} {\ partial (\ lambda x_ {2})}} + \ dots + x_ {n} {\ frac {\ partial f (\ lambda x_ {1}, \ dots, \ lambda x_ {n})} {\ partial (\ lambda x_ {n})}} = {\ frac {\ partial C \ left (\ lambda \ right)} {\ partial \ lambda}} f (x_ {1}, \ dots, x_ {n}).}

We differentiate both sides of the same relation with respect to $x_{k},$ ${\ displaystyle x_ {k},}$ we obtain the relations

\lambda {\frac {\partial f(\lambda x_{1},\dots ,\lambda x_{n})}{\partial (\lambda x_{k})}}=C\left(\lambda \right){\frac {\partial f(x_{1},\dots ,x_{n})}{\partial x_{k}}}.

{\ displaystyle \ lambda {\ frac {\ partial f (\ lambda x_ {1}, \ dots, \ lambda x_ {n})} {\ partial (\ lambda x_ {k})}} = C \ left (\ lambda \ right) {\ frac {\ partial f (x_ {1}, \ dots, x_ {n})} {\ partial x_ {k}}}.}

From here

{\frac {1}{f(x_{1},\dots ,x_{n})}}\left(x_{1}{\frac {\partial f(x_{1},\dots ,x_{n})}{\partial x_{1}}}+\dots +x_{n}{\frac {\partial f(x_{1},\dots ,x_{n})}{\partial x_{n}}}\right)={\frac {\lambda }{C\left(\lambda \right)}}{\frac {\partial C\left(\lambda \right)}{\partial \lambda }}.

{\ displaystyle {\ frac {1} {f (x_ {1}, \ dots, x_ {n})}} left (x_ {1} {\ frac {\ partial f (x_ {1}, \ dots, x_ {n})} {\ partial x_ {1}}} + \ dots + x_ {n} {\ frac {\ partial f (x_ {1}, \ dots, x_ {n})} {\ partial x_ { n}}} \ right) = {\ frac {\ lambda} {C \ left (\ lambda \ right)}} {\ frac {\ partial C \ left (\ lambda \ right)} {\ partial \ lambda}} .}

The right side depends only on $\lambda ,$ ${\ displaystyle \ lambda,}$ the left side depends only on $x_{1},x_{2},\dots ,x_{n}$ ${\ displaystyle x_ {1}, x_ {2}, \ dots, x_ {n}}$ Therefore, they are both equal to the same constant, which we denote by $q .$ ${\ displaystyle q.}$ From the condition ${\frac {\lambda }{C\left(\lambda \right)}}{\frac {\partial C\left(\lambda \right)}{\partial \lambda }}=q$ ${\ displaystyle {\ frac {\ lambda} {C \ left (\ lambda \ right)}} {\ frac {\ partial C \ left (\ lambda \ right)} {\ partial \ lambda}} = q}$ and conditions $C\left(1\right)=1$ ${\ displaystyle C \ left (1 \ right) = 1}$ follows that $C\left(\lambda \right)=\lambda ^{q}.$ ${\ displaystyle C \ left (\ lambda \ right) = \ lambda ^ {q}.}$ Hence, $f\left(x_{1},x_{2},\dots ,x_{n}\right)$ ${\ displaystyle f \ left (x_ {1}, x_ {2}, \ dots, x_ {n} \ right)}$ - homogeneous function with uniformity parameter $q .$ ${\ displaystyle q.}$ Degenerate cases $C\left(\lambda \right)\equiv 0$ ${\ displaystyle C \ left (\ lambda \ right) \ equiv 0}$ and $f\left(x_{1},x_{2},\dots ,x_{n}\right)\equiv 0$ ${\ displaystyle f \ left (x_ {1}, x_ {2}, \ dots, x_ {n} \ right) \ equiv 0}$ are considered separately and are not of interest.

Note. It is not necessary to use the condition $C\left(1\right)=1,$ ${\ displaystyle C \ left (1 \ right) = 1,}$ generally speaking, not initially specified, and also forced to consider a function $C\left(\lambda \right)$ ${\ displaystyle C \ left (\ lambda \ right)}$ out of range $\lambda \in \left[\lambda _{0}-\varepsilon ,\lambda _{0}+\varepsilon \right].$ ${\ displaystyle \ lambda \ in \ left [\ lambda _ {0} - \ varepsilon, \ lambda _ {0} + \ varepsilon \ right].}$ . From equality

{\frac {1}{f}}\left(x_{1}{\frac {\partial f}{\partial x_{1}}}+x_{2}{\frac {\partial f}{\partial x_{2}}}+\dots +x_{n}{\frac {\partial f}{\partial x_{n}}}\right)=q

{\ displaystyle {\ frac {1} {f}} \ left (x_ {1} {\ frac {\ partial f} {\ partial x_ {1}}} + x_ {2} {\ frac {\ partial f} {\ partial x_ {2}}} + \ dots + x_ {n} {\ frac {\ partial f} {\ partial x_ {n}}} \ right) = q}

according to Euler's homogeneous function theorem, it also follows that $f\left(x_{1},x_{2},\dots ,x_{n}\right)$ ${\ displaystyle f \ left (x_ {1}, x_ {2}, \ dots, x_ {n} \ right)}$ - homogeneous function with uniformity parameter $q .$ ${\ displaystyle q.}$ From this, in particular, it follows that if the homogeneity relation is valid for a certain interval $\lambda \in \left[\lambda _{0}-\varepsilon ,\lambda _{0}+\varepsilon \right],$ ${\ displaystyle \ lambda \ in \ left [\ lambda _ {0} - \ varepsilon, \ lambda _ {0} + \ varepsilon \ right],}$ then it is valid for all $\lambda >0.$ ${\ displaystyle \ lambda> 0.}$

2. Let

f\left(\lambda x_{1},\lambda x_{2},\dots ,\lambda x_{n}\right)=Cf\left(x_{1},x_{2},\dots ,x_{n}\right)

{\ displaystyle f \ left (\ lambda x_ {1}, \ lambda x_ {2}, \ dots, \ lambda x_ {n} \ right) = Cf \ left (x_ {1}, x_ {2}, \ dots , x_ {n} \ right)}

at some fixed values $C\neq 0,$ ${\ displaystyle C \ neq 0,}$ $\lambda \neq 1$ ${\ displaystyle \ lambda \ neq 1}$ and arbitrary $x_{1},x_{2},\dots ,x_{n}.$ ${\ displaystyle x_ {1}, x_ {2}, \ dots, x_ {n}.}$ What should be the function $f\left(x_{1},x_{2},\dots ,x_{n}\right)?$ ${\ displaystyle f \ left (x_ {1}, x_ {2}, \ dots, x_ {n} \ right)?}$

Decision. If $x_{1}=0,$ ${\ displaystyle x_ {1} = 0,}$ then the problem is reduced to a functional equation of smaller dimension

f\left(0,\lambda x_{2},\dots ,\lambda x_{n}\right)=Cf\left(0,x_{2},\dots ,x_{n}\right),

{\ displaystyle f \ left (0, \ lambda x_ {2}, \ dots, \ lambda x_ {n} \ right) = Cf \ left (0, x_ {2}, \ dots, x_ {n} \ right) ,}

until it comes down to the case $f\left(0,0,\dots ,0\right)=Cf\left(0,0,\dots ,0\right)$ ${\ displaystyle f \ left (0,0, \ dots, 0 \ right) = Cf \ left (0,0, \ dots, 0 \ right)}$ with an obvious answer $f\left(0,0,\dots ,0\right)=0.$ ${\ displaystyle f \ left (0,0, \ dots, 0 \ right) = 0.}$ Therefore, only the case can be considered below. $x_{1}\neq 0.$ ${\ displaystyle x_ {1} \ neq 0.}$

Let's make a change of variables $x_{1}=y,$ ${\ displaystyle x_ {1} = y,}$ $x_{2}=t_{2}\cdot y,$ ${\ displaystyle x_ {2} = t_ {2} \ cdot y,}$ $x_{3}=t_{3}\cdot y,$ ${\ displaystyle x_ {3} = t_ {3} \ cdot y,}$ $x_{n}=t_{n}\cdot y.$ ${\ displaystyle x_ {n} = t_ {n} \ cdot y.}$ Then $f(x_{1},x_{2},\dots ,x_{n})\to F(y,t_{2},\dots ,t_{n})$ ${\ displaystyle f (x_ {1}, x_ {2}, \ dots, x_ {n}) \ to F (y, t_ {2}, \ dots, t_ {n})}$ and the functional equation takes the form

F\left(\lambda y,t_{2},\dots ,t_{n}\right)=CF\left(y,t_{2},\dots ,t_{n}\right).

{\ displaystyle F \ left (\ lambda y, t_ {2}, \ dots, t_ {n} \ right) = CF \ left (y, t_ {2}, \ dots, t_ {n} \ right).}

Cases should be considered separately. $C>0$ ${\ displaystyle C> 0}$ and $C<0,$ ${\ displaystyle C <0,}$ $\lambda >0$ ${\ displaystyle \ lambda> 0}$ and $\lambda <0,$ ${\ displaystyle \ lambda <0,}$ $y>0$ ${\ displaystyle y> 0}$ and $y<0.$ ${\ displaystyle y <0.}$ Let be $C>0,$ ${\ displaystyle C> 0,}$ $\lambda >0$ ${\ displaystyle \ lambda> 0}$ and $y>0.$ ${\ displaystyle y> 0.}$ Then after the logarithm of both sides of the equality and the replacement $\log y\to t,$ ${\ displaystyle \ log y \ to t,}$ $\log F(y,\dots )\to \Phi (t,\dots )$ ${\ displaystyle \ log F (y, \ dots) \ to \ Phi (t, \ dots)}$ we get the condition

\Phi \left(t+\log \lambda ,\dots \right)=\log C+\Phi \left(t,\dots \right),

{\ displaystyle \ Phi \ left (t + \ log \ lambda, \ dots \ right) = \ log C + \ Phi \ left (t, \ dots \ right),}

whence it follows that $\Phi \left(t,\dots \right)$ ${\ displaystyle \ Phi \ left (t, \ dots \ right)}$ has the form $\Omega \left(t,\dots \right)+{\frac {\log C}{\log \lambda }}t,$ ${\ displaystyle \ Omega \ left (t, \ dots \ right) + {\ frac {\ log C} {\ log \ lambda}} t,}$ Where $\Omega \left(t,\dots \right)$ ${\ displaystyle \ Omega \ left (t, \ dots \ right)}$ - function periodic in variable $t$ ${\ displaystyle t}$ with a period $\log \lambda .$ ${\ displaystyle \ log \ lambda.}$ The converse is obvious: the function

f\left(x_{1},x_{2},\dots ,x_{n}\right)=\Omega \left(\log x_{1},{\frac {x_{2}}{x_{1}}},\dots {\frac {x_{n}}{x_{1}}}\right)\exp \left({\frac {\log C\cdot \log x_{1}}{\log \lambda }}\right),

{\ displaystyle f \ left (x_ {1}, x_ {2}, \ dots, x_ {n} \ right) = \ Omega \ left (\ log x_ {1}, {\ frac {x_ {2}} { x_ {1}}}, \ dots {\ frac {x_ {n}} {x_ {1}}} \ right) \ exp \ left ({\ frac {\ log C \ cdot \ log x_ {1}} { \ log \ lambda}} \ right),}

Where $\Omega \left(t,\dots \right)$ ${\ displaystyle \ Omega \ left (t, \ dots \ right)}$ - function periodic in variable $t$ ${\ displaystyle t}$ with a period $\log \lambda ,$ ${\ displaystyle \ log \ lambda,}$ satisfies the required functional relation for $x_{1}>0.$ ${\ displaystyle x_ {1}> 0.}$

For half shaft $x_{1}<0$ ${\ displaystyle x_ {1} <0}$ replacement used $\log(-y)\to t$ ${\ displaystyle \ log (-y) \ to t}$ and after similar reasoning we get the final answer:

what if

x_{1}>0

{\ displaystyle x_ {1}> 0}

then

f\left(x_{1},x_{2},\dots ,x_{n}\right)=\Omega _{+}\left(\log(+x_{1}),x_{2}/x_{1},\dots x_{n}/x_{1}\right)\exp \left({\frac {\log C\cdot \log(+x_{1})}{\log \lambda }}\right),

{\ displaystyle f \ left (x_ {1}, x_ {2}, \ dots, x_ {n} \ right) = \ Omega _ {+} \ left (\ log (+ x_ {1}), x_ {2 } / x_ {1}, \ dots x_ {n} / x_ {1} \ right) \ exp \ left ({\ frac {\ log C \ cdot \ log (+ x_ {1})} {\ log \ lambda }} \ right),}

b) if

x_{1}<0

{\ displaystyle x_ {1} <0}

then

f\left(x_{1},x_{2},\dots ,x_{n}\right)=\Omega _{-}\left(\log(-x_{1}),x_{2}/x_{1},\dots x_{n}/x_{1}\right)\exp \left({\frac {\log C\cdot \log(-x_{1})}{\log \lambda }}\right),

{\ displaystyle f \ left (x_ {1}, x_ {2}, \ dots, x_ {n} \ right) = \ Omega _ {-} \ left (\ log (-x_ {1}), x_ {2 } / x_ {1}, \ dots x_ {n} / x_ {1} \ right) \ exp \ left ({\ frac {\ log C \ cdot \ log (-x_ {1})} {\ log \ lambda }} \ right),}

or in abbreviated form

f\left(x_{1},x_{2},\dots ,x_{n}\right)=\Omega _{\pm }\left(\log |x_{1}|,{\frac {x_{2}}{x_{1}}},\dots {\frac {x_{n}}{x_{1}}}\right)\exp \left({\frac {\log C\cdot \log |x_{1}|}{\log \lambda }}\right),

{\ displaystyle f \ left (x_ {1}, x_ {2}, \ dots, x_ {n} \ right) = \ Omega _ {\ pm} \ left (\ log | x_ {1} |, {\ frac {x_ {2}} {x_ {1}}}, \ dots {\ frac {x_ {n}} {x_ {1}}} \ right) \ exp \ left ({\ frac {\ log C \ cdot \ log | x_ {1} |} {\ log \ lambda}} \ right),}

where is the designation $\Omega _{\pm }\left(\log |x_{1}|,\dots \right)$ ${\ displaystyle \ Omega _ {\ pm} \ left (\ log | x_ {1} |, \ dots \ right)}$ emphasizes that with $x_{1}>0$ ${\ displaystyle x_ {1}> 0}$ and with $x_{1}<0$ ${\ displaystyle x_ {1} <0}$ these are, generally speaking, two different periodic functions $\Omega _{+}\left(t,\dots \right)$ ${\ displaystyle \ Omega _ {+} \ left (t, \ dots \ right)}$ and $\Omega _{-}\left(t,\dots \right)$ ${\ displaystyle \ Omega _ {-} \ left (t, \ dots \ right)}$ each with a scope $t\in (-\infty ,+\infty )$ ${\ displaystyle t \ in (- \ infty, + \ infty)}$ and different values for this area, but with the same period.

Happening $C<0,$ ${\ displaystyle C <0,}$ $\lambda >0$ ${\ displaystyle \ lambda> 0}$ simplified by the fact that from the chain of relations

F\left(\lambda ^{2}y,t_{2},\dots ,t_{n}\right)=CF\left(\lambda y,t_{2},\dots ,t_{n}\right)=C^{2}F\left(y,t_{2},\dots ,t_{n}\right)

{\ displaystyle F \ left (\ lambda ^ {2} y, t_ {2}, \ dots, t_ {n} \ right) = CF \ left (\ lambda y, t_ {2}, \ dots, t_ {n } \ right) = C ^ {2} F \ left (y, t_ {2}, \ dots, t_ {n} \ right)}

the case already considered by us follows. Therefore function $f\left(x_{1},x_{2},\dots ,x_{n}\right)$ ${\ displaystyle f \ left (x_ {1}, x_ {2}, \ dots, x_ {n} \ right)}$ can be written as

f\left(x_{1},x_{2},\dots ,x_{n}\right)=\Omega _{\pm }\left(\log |x_{1}|,{\frac {x_{2}}{x_{1}}},\dots {\frac {x_{n}}{x_{1}}}\right)\exp \left({\frac {\log |C|\cdot \log |x_{1}|}{\log \lambda }}\right),

{\ displaystyle f \ left (x_ {1}, x_ {2}, \ dots, x_ {n} \ right) = \ Omega _ {\ pm} \ left (\ log | x_ {1} |, {\ frac {x_ {2}} {x_ {1}}}, \ dots {\ frac {x_ {n}} {x_ {1}}} \ right) \ exp \ left ({\ frac {\ log | C | \ cdot \ log | x_ {1} |} {\ log \ lambda}} \ right),}

Where $\Omega _{\pm }\left(t,\dots \right)$ ${\ displaystyle \ Omega _ {\ pm} \ left (t, \ dots \ right)}$ - some function periodic in a variable $t$ ${\ displaystyle t}$ with a period $2\log \lambda .$ ${\ displaystyle 2 \ log \ lambda.}$ Substituting this expression into the original equation shows that $\Omega _{\pm }\left(t,\dots \right)$ ${\ displaystyle \ Omega _ {\ pm} \ left (t, \ dots \ right)}$ - not just a periodic function with a period $2\log \lambda ,$ ${\ displaystyle 2 \ log \ lambda,}$ but anti-periodic with a period $\log \lambda :$ ${\ displaystyle \ log \ lambda:}$

\Omega _{\pm }\left(t+\log \lambda ,\dots \right)=-\Omega _{\pm }\left(t,\dots \right)

{\ displaystyle \ Omega _ {\ pm} \ left (t + \ log \ lambda, \ dots \ right) = - \ Omega _ {\ pm} \ left (t, \ dots \ right)}

(obviously anti-periodicity with period $\log \lambda$ ${\ displaystyle \ log \ lambda}$ entails periodicity with a period $2\log \lambda$ ${\ displaystyle 2 \ log \ lambda}$ ) The converse is obvious: the indicated formula with anti-periodic function $\Omega _{\pm }\left(t,\dots \right)$ ${\ displaystyle \ Omega _ {\ pm} \ left (t, \ dots \ right)}$ satisfies the required functional equation.

Happening $\lambda <0$ ${\ displaystyle \ lambda <0}$ has an additional feature that the half shafts $y<0$ ${\ displaystyle y <0}$ and $y>0$ ${\ displaystyle y> 0}$ affect each other. Consider the case $y>0.$ ${\ displaystyle y> 0.}$ Then from the chain of relations

F\left(\lambda ^{2}y,t_{2},\dots ,t_{n}\right)=CF\left(\lambda y,t_{2},\dots ,t_{n}\right)=C^{2}F\left(y,t_{2},\dots ,t_{n}\right)

{\ displaystyle F \ left (\ lambda ^ {2} y, t_ {2}, \ dots, t_ {n} \ right) = CF \ left (\ lambda y, t_ {2}, \ dots, t_ {n } \ right) = C ^ {2} F \ left (y, t_ {2}, \ dots, t_ {n} \ right)}

it follows that for $x_{1}>0$ ${\ displaystyle x_ {1}> 0}$ function $f\left(x_{1},x_{2},\dots ,x_{n}\right)$ ${\ displaystyle f \ left (x_ {1}, x_ {2}, \ dots, x_ {n} \ right)}$ should look

f\left(x_{1},x_{2},\dots ,x_{n}\right)=\Omega \left(\log |x_{1}|,{\frac {x_{2}}{x_{1}}},\dots {\frac {x_{n}}{x_{1}}}\right)\exp \left({\frac {\log |C|\cdot \log |x_{1}|}{\log |\lambda |}}\right),

Homogeneous function

Alternative definition of a homogeneous function

Properties

Lambda-homogeneous functions

Euler Operator

Limited homogeneous functions

Attached homogeneous functions

Mutually homogeneous functions

Some functional equations related to homogeneous functions

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