全纯函数（holomorphic function）是复分析研究的中心对象；它们是定义在复平面C的开子集上的，在复平面C中取值的，在每点上皆复可微的函数。这是比实可微强得多的条件，暗示着此函数无穷可微并可以用泰勒级数来描述。

解析函数（analytic function）一词经常可以和“全纯函数”互相交换使用，虽然前者有几个其他含义。

全纯函数有时称为正则函数。在整个复平面上都全纯的函数称为整函数（entire function）。「在一点a全纯」不仅表示在a可微，而且表示在某个中心为a的复平面的开邻域上可微。双全纯（biholomorphic）表示一个有全纯逆函数的全纯函数。

In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain. The existence of a complex derivative in a neighborhood is a very strong condition, for it implies that any holomorphic function is actually infinitely differentiable and equal to its own Taylor series.