$\Gamma$函数在教学中的设计策略及其应用分析

Analysis of design strategy and its application of the gamma function in teaching

摘要

$\Gamma$函数作为反常积分中的一个特殊函数,它将阶乘运算推广到实数域和复数域上,是数学教学中不可避开的一个知识点.在教学过程中,对于$\Gamma$函数的讲解仅在反常积分中涉及,鲜有学时来分析其具体性质及其应用,学生在学习过程中也容易忽略$\Gamma$函数的应用.针对此问题,在介绍$\Gamma$函数的相关概念及其性质基础上,重点讨论和分析了$\Gamma$函数在微积分、概率论与数理统计、物理学、工程学和经济学等领域中的具体应用,并结合$\Gamma$函数的特点与应用给出了在教学中的设计策略,最后简要分析了$\Gamma$函数的优势和局限性以及在其他领域的潜在应用.

Abstract

As a special function in the improper integral, the gamma function extends the factorial operation to real number and complex number fields, which is an unavoidable knowledge point in the teaching of mathematics. In teaching process, the explanation of the gamma function is only involved in improper integrals, with little time to analyze its specific properties and applications, and the students are also prone to overlook the application of the gamma function in learning process. To address this issue, on the basis of introducing the relevant concepts of the gamma function and the properties, this paper focused on discussing and analyzing the specific applications of the gamma function in the fields of calculus, probability theory and mathematical statistics, physics, engineering and economics, and proposed the design strategy in teaching based on the characteristics and applications of the gamma function. Finally, the advantages and limitations of the gamma function and potential applications in other fields were briefly analyzed.

关键词

$\Gamma$函数 / $\Gamma$分布 / 应用 / 设计策略 / 潜在应用

Key words

gamma function / gamma distribution / application / design strategy / potential application

参考文献

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基金资助

空军军医大学基础医学院教学研究课题(2022-JCJXKT-YB-15)