Laurent series
Let an agent handle Laurent series by function, centre, annulus of convergence, principal part and residues.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify
Researched by: Claude
Purpose and description
Let an agent handle Laurent series by function, centre, annulus of convergence, principal part and residues.
A representation of a complex function as a power series that includes terms of negative degree, centred at a point, used where a Taylor series does not exist, such as around isolated singularities; truncated forms are Laurent polynomials.
What it is for: Complex analysis, residue calculus and classifying singularities.
It can be compute the expansion; find the annulus of convergence; read off residues; classify singularities.
Distinguishing features
Includes negative powers
Converges on an annulus
Principal part classifies singularities
Generalises Taylor series
What it looks like
A doubly infinite sum of powers of (z - a).
How it is recognised
Negative powers of (z - a)
Principal part
Laurent polynomials are finite sums
Related models
generalises - series
is a kind of - category
is used in - application
is studied in - field
In practice
Families and kinds
Laurent series around poles
Laurent series around essential singularities
Laurent polynomials
formal Laurent series
Failure modes and hazards
Using the wrong annulus
Sign errors in coefficients
Also called
Where this came from
wikidata · CC0 1.0
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.
Expansion Which series.
Function and centre.
Function
What is expanded.
Function
Function and centre.
- Which function is expanded, about which point? definition
- What are the leading terms? measurement
Coefficients
Computation.
Coefficients
Coefficients.
- How are the coefficients computed? action
- Is there a closed form? definition
Convergence Annulus.
Convergence is on an annulus.
Annulus
Radii.
Annulus
Annulus.
- What are the inner and outer radii? measurement
- Which annulus is relevant here? boundary
Uniqueness
Per annulus.
Uniqueness
Uniqueness.
- Is the expansion unique on this annulus? definition
- How do expansions differ across annuli? definition
Singularities Classification.
The principal part classifies.
Principal part
Negative powers.
Principal
Principal part.
- What is the principal part? measurement
- Is the singularity removable, a pole or essential? definition
Residue
Coefficient of 1/(z - a).
Residue
Residue.
- What is the residue? measurement
- How is it used in integrals? action
Learning Teaching.
Laurent series are subtle.
Examples
Worked cases.
Examples
Worked examples.
- Which standard examples illustrate it? definition
- What mistakes are common? boundary
Comparison
With Taylor.
Comparison
Comparison with Taylor series.
- When does a Taylor series suffice? boundary
- When is Laurent needed? boundary
What the second pass must settle
- Should Laurent polynomials be a separate entry?
- How should standard expansions be listed?
- How should formal series be linked?