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Taylor Expansion of homogeneous functions

Joachim Paulusch, Sebastian Schlütter

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.

Taylor Expansion of homogeneous functions

Abstract

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

Paper Structure

This paper contains 2 theorems, 14 equations.

Key Result

Theorem \oldthetheorem

Let $U \subseteq \mathbb{R}^n$ be open and $f:U \to \mathbb{R}$ be positive homogeneousI.e. $f(\lambda x) = \lambda^m f(x)$ for all $x\in U$, $\lambda > 0$ with $\lambda x \in U$. This gives rise to a unique extension of $f$ to the set $\{ \lambda x\,|\, x \in U, \lambda >0\}$. In case $0 \in U$, th

Theorems & Definitions (4)

  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Remark \oldthetheorem
  • proof : Proof of Theorem \ref{['TaylorThm']}