莫里卡瓦问题的代数与解析结构
Algebraic and analytic structure of Morikawa's sangaku problem
AI总结：
本文研究了莫里卡瓦问题的代数与解析结构，证明μ(r)是一个代数函数，并展示了其在特定点的泰勒展开方法。
摘要：
令μ(r)表示由两个半径为1和r≥1的相切圆以及它们的公切线所形成的曲线三角形区域内内接正方形的最小边长。在19世纪初期的日本，莫里卡瓦提出了寻找μ(r)闭式表达式的问题。Holly和Krumm（2021）证明了不存在用根式表示μ(r)的表达式。本文证明μ是一个代数函数，并且在[1,∞)上除了一个可以显式计算的有限集合外，是实解析的。特别地，尽管不存在根式表达式，该函数在所有非异常值的r处都允许收敛的泰勒展开，其系数可通过从定义的代数方程中进行牛顿迭代来计算。我们通过显式计算μ(r)在r=1处的泰勒展开来说明该方法。
英文摘要：
Let $μ(r)$ denote the minimal side length of a square inscribed in the curvilinear triangular region formed by two tangent circles of radii $1$ and $r \ge 1$ together with their common tangent line. The problem of finding a closed-form expression for $μ(r)$ was posed in early nineteenth-century Japan by Morikawa. It was proved by Holly and Krumm (2021) that no expression in radicals exists for $μ(r)$. In this article we show that $μ$ is an algebraic function, and consequently real-analytic on $[1,\infty)$ outside a finite explicitly computable set. In particular, although no expression in radicals exists, the function admits convergent Taylor expansions at all non-exceptional values of $r$, whose coefficients may be computed by Newton iteration from the defining algebraic equation. We illustrate the method by explicitly computing the Taylor expansion of $μ(r)$ centered at $r=1$.
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