Certain identities on derivatives of radial homogeneous and logarithmic functions

TL;DR

This paper extends known identities for derivatives of radial homogeneous and logarithmic functions from one dimension to higher dimensions by introducing a suitable derivative norm and calculating exact multiples.

Contribution

It generalizes derivative identities of radial functions to higher dimensions with explicit formulas for the multiples involved.

Findings

Derived exact values of derivative multiples in higher dimensions.

Extended 1D identities to multi-dimensional radial functions.

Provided explicit formulas for derivatives of logarithmic functions.

Abstract

Let $k$ be a natural number and $s$ be real. In the 1-dimensional case, the $k$-th order derivatives of the functions $\lvert x\rvert^s$ and $\log \lvert x\rvert$ are multiples of $\lvert x\rvert^{s-k}$ and $\lvert x\rvert^{-k}$, respectively. In the present paper, we generalize this fact to higher dimensions by introducing a suitable norm of the derivatives, and give the exact values of the multiples.