Homogeneous function: Difference between revisions

Revision as of 20:35, 15 March 2022

Revision as of 03:59, 5 July 2022

Line 43:

This change allow considering (positively) homogeneous functions with any real number as their degrees, since [[exponentiation]] with a positive real base is well defined.

Even in the case of integer degrees, there are many useful functions that are positively homogeneous without being homogeneous. This is, in particular, the case of the [[absolute value]] function and [[norm (mathematics)|norms]], which are all positively homogeneous of degree {{math|1}}. They are not homogeneous since <math>|-x|=-|x|\neq -|x|</math> if <math>x\neq 0.</math> This remains true in the [[complex number|complex]] case, since the field of the complex numbers <math>\C</math> and every complex vector space can be considered as real vector spaces.

[[#Euler's theorem|Euler's homogeneous function theorem]] is a characterization of positively homogeneous [[differentiable function]]s, which may be considered as the ''fundamental theorem on homogeneous functions''.