complex contour integral

复积分

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

Translation evidence: Literature usage

Meaning in context

For continuous f along a piecewise C^1 path gamma:[a,b] to C, its contour integral is integral_a^b f(gamma(t)) gamma'(t) dt, summed over the smooth pieces.

沿分段一阶光滑路径的连续复函数，其复积分由参数式 f(γ(t))γ'(t) 对参数积分定义，并对各光滑段求和。

Example

Along γ(t)=2eit\gamma(t)=2e^{it}, 0≤t≤2π0\le t\le2\pi, ∫dz/z=∫02πi dt=2πi\int dz/z=\int_0^{2\pi}i\,dt=2\pi i.

Usage and distinctions

The factor gamma' includes orientation; this differs from arc-length integration against |gamma'|.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Michael Taylor — Introduction to Complex Analysis§1, equations (1.44) and (1.44B), printed p.30 · Checked 2026-10-14

Translation sources

  1. 肇庆学院 —《复变函数》课程教学大纲第三章，第一节 复积分概念及其简单性质，printed p.6 · Checked 2026-10-14
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