# An entire function connected with the approximation of the golden ratio

## TL;DR

The paper studies a special entire function related to the approximation of the golden ratio, revealing its complex structure and connection to nested radicals and fractal zero sets.

## Contribution

It introduces an entire inverse function of a non-entire function connected to nested radicals approximating the golden ratio, and explores its fractal zero structure.

## Key findings

- The inverse function is entire and satisfies Poincare equality.
- Zeros of the inverse function form fractal structures similar to Julia sets.
- The original function relates to the convergence of nested radicals towards the golden ratio.

## Abstract

In 1987, R. B. Paris uses the analytic function \[\label{main} g(w)=\lim_{n\to\infty}(2\varphi)^n\biggl(\underbrace{\sqrt{1+\sqrt{1+...\sqrt{1+w}}}}_n-\varphi\biggr),\ \ \ \varphi=\frac{1+\sqrt{5}}2, \] to estimate the convergence of nested squares to the golden ratio. The function $g$ is non-entire and, perhaps, can not be expressed in terms of some standard known functions. We show that $f(z):=g^{-1}(z)$ is an entire function satisfying Poincare equality. While $f$ has zeros of various multiplicities, it can be expressed in terms of its simple zeros, forming fractal structures similar to Julia sets.