关于根式函数的多值性探讨

On The Multi-valuedness of Root Function

复变函数已经被广泛地应用在理论物理、弹性理论和天体力学等方面,而且与数学中其他分支的联系也日益密切。复变函数的主要特点就是多值性。多值函数在G平面上可以分解成单值解析分支,然而关于G平面的概念教材上解释的很模糊,现有的文献也很少谈及,即使谈及的,也将G平面狭义地理解为主值支的定义域。重新阐述了G平面的定义,剖析了平面的结构,并在G平面上研究了根式函数的单值解析分支。

The complex variable function has been widely used in theoretical physics, theory of elasticity, celestial mechanics andother disciplines. Also, it has an increasingly close contact with other branches of mathematics. The main characteristic of complexfunction is multivalued. The multi-valued function can be decomposed into a single value analytic branch on the G plane. However, thedefinition of G plane is unintelligible in the textbook, and rarely talked about by the existing literature. Even when the G plane is talkedabout, it’s only regarded as the definition domain of the main value branch. This paper reinterprets the definition and analyzes thestructure of G plane, and studies the single-valued analytic branch of root function on the G plane.