holomorphic function

全纯函数

Analysis, Measure & PDE · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

A complex-valued function on an open subset of C is holomorphic when its complex difference quotient has a finite limit at every point, with the increment approaching zero through all complex directions.

复平面开集上的复值函数，若每一点的复增量商沿任意方向趋于零时都有同一有限极限，则为全纯函数。

Example

For f(z)=z3f(z)=z^3, the quotient is 3z2+3zh+h2→3z23z^2+3zh+h^2\to3z^2.

Usage and distinctions

Real differentiability of a map R^2 to R^2 is weaker; complex differentiability requires a complex-linear differential.

Read the encyclopedia explanation

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Michael Taylor — Introduction to Complex Analysis§1, definition after (1.4); §8, Goursat's theorem · Checked 2026-10-14

Translation sources

  1. 肇庆学院 —《复变函数》课程教学大纲第二章，学习重点及难点：初等全纯函数；第四章：全纯函数的Taylor展开，printed pp.5,8 · Checked 2026-10-14
Edit source / propose a PRReport a correction

Original encyclopedia and terminology content: CC BY-SA 4.0. Contribute or report a correction.