--- title: "Laurent Series and Classifying Isolated Singularities" description: "Allow negative powers to expand a function on an annulus, and sort singularities by how the negative part behaves." canonical: https://lightmysky.com/learn/mathematics/laurent-series-and-classifying-isolated-singularities-mt_WoQV0gD1O- source: https://lightmysky.com/learn/mathematics/laurent-series-and-classifying-isolated-singularities-mt_WoQV0gD1O-.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Laurent Series and Classifying Isolated Singularities Allow negative powers to expand a function on an annulus, and sort singularities by how the negative part behaves. Subject: Mathematics · Area: Complex Analysis · Ages 21 to 22 Page: https://lightmysky.com/learn/mathematics/laurent-series-and-classifying-isolated-singularities-mt_WoQV0gD1O- ## Ready when they can - Find the Laurent expansion of a function on a stated annulus - Classify a singularity as removable, a pole of stated order, or essential - Give the behaviour near an essential singularity that separates it from a pole ## Lesson: Negative powers name the hole Laurent series. A function holomorphic on an annulus r < |z - a| < R expands as a sum over all integers n of c_n (z - a)^n, converging locally uniformly there. The centre and annulus are part of the specification; the same function expands differently on different annuli. A missing or problem point a is an isolated singularity when f is holomorphic on some punctured disk 0 < |z - a| < R. If f is already holomorphic at a, the point is regular, not a singularity. **Example.** Worked two-region expansion. For f(z) = 1/(z - 1) about 0, on |z| < 1 write -1/(1 - z) and expand as -sum_{n>=0} z^n, with no negative powers. On |z| > 1 write (1/z)/(1 - 1/z) and expand as sum_{n>=1} z^(-n), all negative powers. The ratio selection decides which powers appear; each expansion is valid only on its stated annulus. Classifying isolated singularities. On a punctured disk 0 < |z - a| < R, read the principal part, the negative powers with nonzero coefficients after combining like powers. No negative powers means removable: the limit fills the point. Finitely many nonzero negative powers means a pole of positive order m, where the deepest nonzero term is (z - a)^(-m), not the term count: 7/(z - a)^5 + 2/(z - a) has order 5. Infinitely many nonzero negative powers means essential. The classification needs the expansion reaching arbitrarily close to a, so an outer annulus cannot classify a point. The coefficient c_-1 is the local residue. Pole versus essential behaviour. Near a pole of order m, |f(z)| tends to infinity as z approaches a, uniformly in every direction. Near an essential singularity, the Casorati picture says the image of every punctured neighbourhood is dense in the complex plane: values land arbitrarily close to every finite complex number, without necessarily attaining them. e^(1/z) never equals 0, yet approaches 0 along the negative real axis and grows along the positive axis. **Example.** Worked essential expansion. The exponential series e^w = sum_{n>=0} w^n/n! holds for every complex w. Multiplying e^w by e^(-w) gives 1, so the exponential is never zero. Substitute w = 1/z: e^(1/z) = sum_{n>=0} 1/(n! z^n) = 1 + 1/z + 1/(2z²) + 1/(6z³) + ..., valid for z nonzero. The n = 0 term is the constant 1, and every later term is a negative power with nonzero coefficient, running without end. The infinitely many nonzero negative powers mark an essential singularity at 0. Removable singularities. If f is holomorphic and bounded on a punctured disk 0 < |z - a| < R, the singularity is removable: the finite limit as z approaches a fills the point and extends f analytically. The constant coefficient c_0 of the Laurent expansion becomes the filled value. Boundedness alone without punctured holomorphicity does not give the extension, as conj(z)/z shows: modulus 1, but values 1 on the real axis and -1 on the imaginary axis. **Recap.** Negative powers name the hole, the principal part fingerprints the type, and density of values separates essential from pole. ## Practice 17 questions on this page, each with its working shown. ## Needs first - [Power Series and Analyticity in the Complex Plane](https://lightmysky.com/learn/mathematics/power-series-and-analyticity-in-the-complex-plane-mt_Mwxapy_pU8) ## Opens up - [The Residue Theorem](https://lightmysky.com/learn/mathematics/the-residue-theorem-mt_BN2biZ_zos)