# An entire free holomorphic function which is unbounded on the row ball

## TL;DR

This paper constructs a specific free holomorphic function that is entire and unbounded on the row ball, demonstrating a notable property in free noncommutative function theory.

## Contribution

It provides the first example of an entire free holomorphic function that is unbounded on the row ball, expanding understanding of free function behavior.

## Key findings

- Constructed an explicit unbounded free holomorphic function on the row ball
- Showed the function is continuous in the free topology
- Demonstrated unboundedness on the set of row contractions

## Abstract

We give an entire free holomorphic function $f$ which is unbounded on the row ball. That is, we give a holomorphic free noncommutative function which is continuous in the free topology developed by Agler and McCarthy but is unbounded on the set of row contractions.