essential isolated singularity

本性奇点

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

Translation evidence: Literature usage

Meaning in context

An isolated singularity of a holomorphic function is essential if it is neither removable nor a pole; equivalently its Laurent expansion has infinitely many nonzero negative-power coefficients.

全纯函数的孤立奇点若既不可去又不是极点，则为本性奇点；等价地，洛朗展开有无穷多个非零负幂项。

Example

e2/z=∑n=0∞2n/(n!zn)e^{2/z}=\sum_{n=0}^\infty 2^n/(n!z^n) has an essential singularity at 00.

Usage and distinctions

The classification requires an isolated singularity; a boundary accumulation of singularities is not classified by this trichotomy.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Michael Taylor — Introduction to Complex Analysis§11, definition after Proposition 11.2, printed p.127; §12, Proposition 12.2 · Checked 2026-10-14

Translation sources

  1. 肇庆学院 —《复变函数》课程教学大纲第五章，第二节 解析函数的孤立奇点，本性奇点，printed p.9 · Checked 2026-10-14
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