Cauchy integral formula

柯西积分公式

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

Translation evidence: Literature usage

Meaning in context

If f is holomorphic near the closed disk of radius r centered at a, then for every z in that disk's interior, f(z)=(2 pi i)^(-1) integral_{|w-a|=r} f(w)/(w-z) dw with counterclockwise orientation.

函数在闭圆盘附近全纯时，其内点值等于沿逆时针边界积分 f(w)/(w-z)，再除以 2πi。

Example

On $|w|=2$, $\int w^2/(w-1)\,dw=2\pi i$ because $f(1)=1$.

Usage and distinctions

The evaluation point must be inside and the function holomorphic throughout the disk; enclosing other singularities invalidates this form.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Michael Taylor — Introduction to Complex Analysis§5, Theorem 5.3 and equations (5.5)–(5.9) · Checked 2026-10-14

Translation sources

1. 肇庆学院 —《复变函数》课程教学大纲第三章，第三节 柯西积分公式及其推论，printed p.7 · Checked 2026-10-14

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