摘要
不考虑连续性的多值运算和考虑连续性的多值函数之间虽有很密切的关系，但两者并不相同．前者虽用处较少但不能省略也不能用后者替代．先讨论多值运算的相等性及其叠合，其单值支及主值支，纠正人们认为在复变元情况不能规定主值的说法．其次引入连续性从而过渡到多值函数，利用分支点与切割线概念而做出全值线与黎曼面，从而它可以及早地使用．我们把对多值函数的处理办法归结为多值函数原则，即在求多值函数值时应遵循使多值项对最内的变元为连续的原则而在不同的适当的单值支上取值，不限于取相同的单值支，由它而统一地讨论实变元与复变元的多值函数，无须象目前的理论那样必须分别处理．最后具体地就实变元情况而讨论新的多值函数理论，并与传统理论相比较，阐明新理论的优越性．
Although the multivalued operations (on which we do not consider the continuity) are closely connected with the multivalued functions ( on which we make use of continuity gravely), yet they are not identical with one another. The former are much less in use, yet they are indispensable and could not be substituted by the latter and hence they are not negligible. In this paper we discuss them both in full. First, we discuss the equality and compositions of multivalued operations. We consider the monodromy branches and the principal value branches. Second, we introduce the continuity and transit to the multivalued functions. We find that for single real variable case, the monodromy branches are in the main continuous, and they may be connected into a single continuous curve, called wholevalue curve; for the case of several real variables or complex variables, there appear several curves of discontinuity, called cuts. Anyhow we may connect them into Riemann surface. We discuss the Riemann surface in quite an elementary fashion. We based the whole theory of multivalued functions on a single principlemultivalued function principle. This principle applies not only to the basic functions but also to arbitrary multivalued terms; not only to the case of complex variables but also to the case of real variables. Therefore, we have a coherent theory of multivalued functions, in contrast to the traditional theory, in which we should treat the two cases (complex variables and real variables) separately. At the last, we discuss concretely the cases of one or several real variables by examples, and show the advantages of our new theory.
出处
《南京大学学报（自然科学版）》
CAS
CSCD
1998年第1期1-2,共2页
Journal of Nanjing University（Natural Science）
关键词
多值函数
黎曼面
单值支
多值运算
multivalued functions
Riemann surfaces
monodromy branch
