Abstract In 1987, R. B. Paris introduced an analytic function g to estimate the rate of convergence of nested square root radicals to the golden ratio. The function g is nonentire and, perhaps, cannot be expressed in terms of some standard known functions. We show that the inverse is an entire function satisfying the Poincaré equality. We provide some explicit expansions of f based on exact formulas for its zeros. The zeros of f form fractal structures similar to Fatou and Julia sets.

中文翻译：

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与黄金比例近似相关的整个函数

摘要 1987 年，RB Paris 引入了一个解析函数 g 来估计嵌套平方根根对黄金分割率的收敛速度。函数 g 是不完整的，也许不能用一些标准的已知函数来表达。我们证明逆是满足庞加莱等式的完整函数。我们根据 f 的零点的精确公式提供了 f 的一些显式扩展。f 的零点形成类似于 Fatou 和 Julia 集的分形结构。