Taylor Expansion of homogeneous functions

Abstract

We derive the Taylor polynomial of a function, which is $m$-times continuously differentiable and positive homogeneous of order $m$. The Taylor polynomial in $a$ for $f(b)$ of order $m$ in general is a polynomial of order $m$ in $b-a$. If the given function is positive homogeneous of order $m$, the Taylor polynomial is a polynomial in $b$ rather than $b-a$, and the order of all terms is $m$. The result can be applied to powers of homogeneous functions of order $1$ as well.