# Cauchy-Riemann equations for free noncommutative functions

## TL;DR

This paper extends the classical Cauchy-Riemann equations to free noncommutative functions on matrix tuples, establishing a link between analyticity and differentiability in a noncommutative setting.

## Contribution

It generalizes the classical Cauchy-Riemann equations to free noncommutative functions and shows that real noncommutative functions are indeed noncommutative functions.

## Key findings

- Extension of Cauchy-Riemann equations to noncommutative functions
- Real noncommutative functions are noncommutative functions
- Framework applicable to matrices of arbitrary size

## Abstract

In classical complex analysis analyticity of a complex function $f$ is equivalent to differentiability of its real and imaginary parts $u$ and $v$, respectively, together with the Cauchy-Riemann equations for the partial derivatives of $u$ and $v$. We extend this result to the context of free noncommutative functions on tuples of matrices of arbitrary size. In this context, the real and imaginary parts become so called real noncommutative functions, as appeared recently in the context of L\"owner's theorem in several noncommutative variables. Additionally, as part of our investigation of real noncommutative functions, we show that real noncommutative functions are in fact noncommutative functions.