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''of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h0476702.png" />''
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$#C+1 = 39 : ~/encyclopedia/old_files/data/H047/H.0407670 Homogeneous function
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A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h0476703.png" /> such that for all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h0476704.png" /> in its domain of definition and all real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h0476705.png" />, the equation
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{{TEX|auto}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h0476706.png" /></td> </tr></table>
+
''of degree  $  \lambda $''
  
holds, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h0476707.png" /> is a real number; here it is assumed that for every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h0476708.png" /> in the domain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h0476709.png" />, the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767010.png" /> also belongs to this domain for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767011.png" />. If
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767012.png" /></td> </tr></table>
+
$$
 +
f ( t x _ {1} \dots t x _ {n} )  = \
 +
t  ^  \lambda  f ( x _ {1} \dots x _ {n} )
 +
$$
  
that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767013.png" /> is a polynomial of degree not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767014.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767015.png" /> is a homogeneous function of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767016.png" /> if and only if all the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767017.png" /> are zero for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767018.png" />. The concept of a homogeneous function can be extended to polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767019.png" /> variables over an arbitrary commutative ring with an identity.
+
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
  
Suppose that the domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767020.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767021.png" /> lies in the first quadrant, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767022.png" />, and contains the whole ray <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767024.png" />, whenever it contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767025.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767026.png" /> is homogeneous of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767027.png" /> if and only if there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767028.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767029.png" /> variables, defined on the set of points of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767030.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767031.png" />, such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767032.png" />,
+
$$
 +
f ( x _ {1} \dots x _ {n} )  = \
 +
\sum _ {0 \leq  k _ {1} + \dots + k _ {n} \leq  m }
 +
a _ {k _ {1}  \dots k _ {n} }
 +
x _ {1} ^ {k _ {1} } \dots x _ {n} ^ {k _ {n} } ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767033.png" /></td> </tr></table>
+
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.
  
If the domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767034.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767035.png" /> is an open set and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767036.png" /> is continuously differentiable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767037.png" />, then the function is homogeneous of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767038.png" /> if and only if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767039.png" /> in its domain of definition it satisfies the Euler formula
+
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 $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047670/h04767040.png" /></td> </tr></table>
+
$$
 +
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
 +
 
 +
$$
 +
\sum _ { i= } 1 ^ { n }
 +
x _ {i}
 +
\frac{\partial  f ( x _ {1} \dots x _ {n} ) }{\partial  x _ {i} }
 +
  = \
 +
\lambda f ( x _ {1} \dots x _ {n} ) .
 +
$$

Revision as of 22:10, 5 June 2020


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=47253
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article