L’ Hospital’ s rule for complex functions and its applications from the perspective of educational mathematics
教育数学视角下复变函数的洛必达法则及其应用

摘要

微积分中的洛必达法则是计算实函数的极限及证明Taylor公式的重要工具，但其证明依赖于Cauchy中值定理，而该定理对复变函数并不成立，故无法直接平行推广。本文在教育数学视角下，利用解析函数的级数展开建立相应的洛必达法则并给出完整证明。该理论成果将经典微积分中的洛必达法则推广至复分析领域，为计算解析函数的极限提供了有效工具；同时，其证明路径以核心命题串联解析性、级数展开、零点与极点等主干知识，显著降低认知门槛，对优化复变函数教学、促进学习者从套用结论到理解本质的转变具有重要教育价值。

Abstract

The L’ Hospital’ s rule in calculus is an important tool for calculating the limits of real functions and proving Taylor’s formula. However, its proof relies on the Cauchy Mean Value Theorem, while does not hold for complex functions. Therefore, it cannot be directly extended in a parallel manner. From the perspective of educational mathematics, this paper uses the series expansions of analytic functions to establish the corresponding L’ Hospital’ s rule and provides a complete proof. This theoretical achievement extends the L’ Hospital’ s rule from classical calculus to the field of complex analysis, which provides an effective tool for calculating the limits of analytic functions. Simultaneously, its proof pathway, which weaves together core knowledge domains including analyticity, series expansions, zeros and poles via a central proposition, notably reduces the cognitive threshold. This bears significant educational value for optimizing the teaching of complex functions and facilitating learners' shift from the mere application of conclusions to the comprehension of their essence.