Laurent series

洛朗展式

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

Translation evidence: Literature usage

Meaning in context

A Laurent series centered at a is a two-sided power expansion sum_{k in Z} c_k(z-a)^k, with its negative- and nonnegative-power parts converging on a specified annulus.

洛朗展式是包含正负整数次幂的双边展开式，其正幂部分和负幂部分须在指定圆环内分别收敛。

Example

For $0<|z|<2$, $1/[z(2-z)]=1/(2z)+\sum_{n=0}^\infty z^n/2^{n+2}$.

Usage and distinctions

Negative powers are allowed; a Taylor expansion around a requires holomorphic extension through its center.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Michael Taylor — Introduction to Complex Analysis§12, Proposition 12.1 and equations (12.1)–(12.8) · Checked 2026-10-14

Translation sources

1. 肇庆学院 —《复变函数》课程教学大纲第五章，第一节 解析函数的洛朗展式，printed p.9 · Checked 2026-10-14

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