关于复域多值函数的复合

由于复域中的对数函数是多值函数,故复域中的对数函数与实域中的对数函数有很多不同的性质,本文首先举出对数函数在实域中的3个代数性质,探讨对数函数在实域和复域中的异同。一般教材中不涉及多值函数的复合问题,利用对数函数与幂函数这2个多值函数的复合,证明Lnzα≠αLn z,并得出关于复域多值函数复合的有关性质。

The Composition of Multi-valued Functions in the Complex Domain

Logarithm function is multi-valued function in the complex domain,so logarithm function in the complex domain and real domain have a lot of differences in properties,this paper firstly give its three algebraic properties in the real domain,and discuss the similarities and differences in both complex and real domain.Now no a textbook involves researching compound multi-valued function,here we use the compound of power function and logarithm function to prove Ln z^α≠αLn z,and draw on the properties of composite function in the complex domain.