The Gateaux Derivative of Map over Division Ring

TL;DR

This paper explores the Gateaux derivative of functions over division rings, including quaternions, deriving Taylor series and solving differential equations, highlighting differences from complex analysis such as the absence of Cauchy-Riemann equations.

Contribution

It extends the concept of derivatives to division rings like quaternions, develops higher-order Gateaux derivatives, and applies these to differential equations and algebraic structures.

Findings

Derived the Gateaux derivative for functions over division rings.

Established Taylor series expansion in non-commutative algebra.

Solved differential equations using the developed framework.

Abstract

I consider differential of mapping $f$ of continuous division ring as linear mapping the most close to mapping $f$. Different expressions which correspond to known deffinition of derivative are supplementary. I explore the Gateaux derivative of higher order and Taylor series. The Taylor series allow solving of simple differential equations. As an example of solution of differential equation I considered a model of exponent. I considered application of obtained theorems to complex field and quaternion algebra. In contrast to complex field in quaternion algebra congugation is linear function of original number \bar a=a+iai+jaj+kak . In quaternion algebra this difference leads to the absence of analogue of the Cauchy Riemann equations that are well known in the theory of complex function.