复变函数的定义

如果存在映射关系 ，把一个复数 映射为一个或者多个复数 ，那么称 为复变函数，记作：


如果一个对应一个那么就是单值函数，如果一个对应多个，那就称为多值函数。

定义一些初等函数

指数函数
三角函数
双曲三角函数
对数函数定义为指数函数的反函数 (注意。该函数是一个多值函数，又是幅角可以转圈)
幂函数

复变函数的导数 重要

(1.20)
设函数为单值函数，当在邻域内按照各种方向，各种方式趋近于，如果极限值存在且相等，即极限（1.20）存在且相等，那么称为在点 可导，极限式子1.20称为在的导数或者微商，记为

柯西—黎曼条件(C-R条件）

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:::info 定理
若存在且连续，那么函数可导的充要条件是C-R条件 :::

一些讨论

• 极坐标下的C-R条件
• 除了多值函数外，初等函数几乎处处可导。
• 复变函数的定义为一切方向、一切路径趋近，比实数函数严格的多，所以，能够导出更多的定理，因为可利用的条件多。重要