Difference between revisions of "Homogeneous function"
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| + | $#C+1 = 39 : ~/encyclopedia/old_files/data/H047/H.0407670 Homogeneous function | ||
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| − | + | ''of degree $ \lambda $'' | |
| − | + | A function $ f $ | |
| + | such that for all points $ ( x _ {1} \dots x _ {n} ) $ | ||
| + | in its domain of definition and all real $ t > 0 $, | ||
| + | the equation | ||
| − | + | $$ | |
| + | f ( t x _ {1} \dots t x _ {n} ) = \ | ||
| + | t ^ \lambda f ( x _ {1} \dots x _ {n} ) | ||
| + | $$ | ||
| − | + | holds, where $ \lambda $ | |
| + | is a real number; here it is assumed that for every point $ ( x _ {1} \dots x _ {n} ) $ | ||
| + | in the domain of $ f $, | ||
| + | the point $ ( t x _ {1} \dots t x _ {n} ) $ | ||
| + | also belongs to this domain for any $ t > 0 $. | ||
| + | If | ||
| − | + | $$ | |
| + | f ( x _ {1} \dots x _ {n} ) = \ | ||
| + | \sum _ {0 \leq k _ {1} + \dots + k _ {n} \leq m } | ||
| + | a _ {k _ {1} \dots k _ {n} } | ||
| + | x _ {1} ^ {k _ {1} } \dots x _ {n} ^ {k _ {n} } , | ||
| + | $$ | ||
| − | < | + | that is, $ f $ |
| + | is a polynomial of degree not exceeding $ m $, | ||
| + | then $ f $ | ||
| + | is a homogeneous function of degree $ m $ | ||
| + | if and only if all the coefficients $ a _ {k _ {1} \dots k _ {n} } $ | ||
| + | are zero for $ k _ {1} + \dots + k _ {n} < m $. | ||
| + | The concept of a homogeneous function can be extended to polynomials in $ n $ | ||
| + | variables over an arbitrary commutative ring with an identity. | ||
| − | + | Suppose that the domain of definition $ E $ | |
| + | of $ f $ | ||
| + | lies in the first quadrant, $ x _ {1} > 0 \dots x _ {n} > 0 $, | ||
| + | and contains the whole ray $ ( t x _ {1} \dots t x _ {n} ) $, | ||
| + | $ t > 0 $, | ||
| + | whenever it contains $ ( x _ {1} \dots x _ {n} ) $. | ||
| + | Then $ f $ | ||
| + | is homogeneous of degree $ \lambda $ | ||
| + | if and only if there exists a function $ \phi $ | ||
| + | of $ n- 1 $ | ||
| + | variables, defined on the set of points of the form $ ( x _ {2} / x _ {1} \dots x _ {n} / x _ {1} ) $ | ||
| + | where $ ( x _ {1} \dots x _ {n} ) \in E $, | ||
| + | such that for all $ ( x _ {1} \dots x _ {n} ) \in E $, | ||
| − | + | $$ | |
| + | f ( x _ {1} \dots x _ {n} ) = \ | ||
| + | x _ {1} ^ \lambda \phi | ||
| + | \left ( | ||
| + | |||
| + | { | ||
| + | \frac{x _ 2}{x _ 1} | ||
| + | } | ||
| + | \dots | ||
| + | |||
| + | { | ||
| + | \frac{x _ n}{x _ 1} | ||
| + | } | ||
| + | \right ) . | ||
| + | $$ | ||
| + | |||
| + | If the domain of definition $ E $ | ||
| + | of $ f $ | ||
| + | is an open set and $ f $ | ||
| + | is continuously differentiable on $ E $, | ||
| + | then the function is homogeneous of degree $ \lambda $ | ||
| + | if and only if for all $ ( x _ {1} \dots x _ {n} ) $ | ||
| + | in its domain of definition it satisfies the Euler formula | ||
| + | |||
| + | $$ | ||
| + | \sum _ { i=1 } ^ { n } | ||
| + | x _ {i} | ||
| + | \frac{\partial f ( x _ {1} \dots x _ {n} ) }{\partial x _ {i} } | ||
| + | = \ | ||
| + | \lambda f ( x _ {1} \dots x _ {n} ) . | ||
| + | $$ | ||
Latest revision as of 16:12, 1 August 2021
of degree $ \lambda $
A function $ f $ such that for all points $ ( x _ {1} \dots x _ {n} ) $ in its domain of definition and all real $ t > 0 $, the equation
$$ f ( t x _ {1} \dots t x _ {n} ) = \ t ^ \lambda f ( x _ {1} \dots x _ {n} ) $$
holds, where $ \lambda $ is a real number; here it is assumed that for every point $ ( x _ {1} \dots x _ {n} ) $ in the domain of $ f $, the point $ ( t x _ {1} \dots t x _ {n} ) $ also belongs to this domain for any $ t > 0 $. If
$$ f ( x _ {1} \dots x _ {n} ) = \ \sum _ {0 \leq k _ {1} + \dots + k _ {n} \leq m } a _ {k _ {1} \dots k _ {n} } x _ {1} ^ {k _ {1} } \dots x _ {n} ^ {k _ {n} } , $$
that is, $ f $ is a polynomial of degree not exceeding $ m $, then $ f $ is a homogeneous function of degree $ m $ if and only if all the coefficients $ a _ {k _ {1} \dots k _ {n} } $ are zero for $ k _ {1} + \dots + k _ {n} < m $. The concept of a homogeneous function can be extended to polynomials in $ n $ variables over an arbitrary commutative ring with an identity.
Suppose that the domain of definition $ E $ of $ f $ lies in the first quadrant, $ x _ {1} > 0 \dots x _ {n} > 0 $, and contains the whole ray $ ( t x _ {1} \dots t x _ {n} ) $, $ t > 0 $, whenever it contains $ ( x _ {1} \dots x _ {n} ) $. Then $ f $ is homogeneous of degree $ \lambda $ if and only if there exists a function $ \phi $ of $ n- 1 $ variables, defined on the set of points of the form $ ( x _ {2} / x _ {1} \dots x _ {n} / x _ {1} ) $ where $ ( x _ {1} \dots x _ {n} ) \in E $, such that for all $ ( x _ {1} \dots x _ {n} ) \in E $,
$$ f ( x _ {1} \dots x _ {n} ) = \ x _ {1} ^ \lambda \phi \left ( { \frac{x _ 2}{x _ 1} } \dots { \frac{x _ n}{x _ 1} } \right ) . $$
If the domain of definition $ E $ of $ f $ is an open set and $ f $ is continuously differentiable on $ E $, then the function is homogeneous of degree $ \lambda $ if and only if for all $ ( x _ {1} \dots x _ {n} ) $ in its domain of definition it satisfies the Euler formula
$$ \sum _ { i=1 } ^ { n } x _ {i} \frac{\partial f ( x _ {1} \dots x _ {n} ) }{\partial x _ {i} } = \ \lambda f ( x _ {1} \dots x _ {n} ) . $$
How to Cite This Entry:
Homogeneous function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homogeneous_function&oldid=11366
Homogeneous function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homogeneous_function&oldid=11366
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article