Math 2.0

Functions of a complex variable Theorem

Existence of Laurent expansions on an annulus

Builds on AnnulusLaurent seriesHolomorphic functionCircle integral

Statement

Let c∈ℂc \in \mathbb{C}, let 0≤r<R≤∞0 \le r < R \le \infty, and let A={z:r<∣z−c∣<R}A = \{z : r < |z - c| < R\} be an annulus. Let f:A→ℂf : A \to \mathbb{C} be holomorphic on AA. Then there are unique complex numbers (an)n∈ℤ(a_n)_{n \in \mathbb{Z}} such that the Laurent series ∑n∈ℤan(z−c)n\sum_{n \in \mathbb{Z}} a_n (z - c)^n converges to f(z)f(z) for each z∈Az \in A. For each ρ\rho with r<ρ<Rr < \rho < R and each n∈ℤn \in \mathbb{Z}, an=12πi∮∣z−c∣=ρf(z)(z−c)n+1 dz.a_n = \frac{1}{2 \pi i} \oint_{|z - c| = \rho} \frac{f(z)}{(z - c)^{n+1}} \, dz. The series converges absolutely on AA and uniformly on each closed annulus ρ1≤∣z−c∣≤ρ2\rho_1 \le |z - c| \le \rho_2 with r<ρ1≤ρ2<Rr < \rho_1 \le \rho_2 < R.

Remarks

Uniqueness means: each Laurent series that converges to ff on AA has these coefficients. So we can find them by any method, for example with known power series. For example, on 0<∣z∣<10 < |z| < 1, 1z(1−z)=1z+1+z+z2+⋯\frac{1}{z(1 - z)} = \frac{1}{z} + 1 + z + z^2 + \cdots. On ∣z∣>1|z| > 1, the same function is −1z2−1z3−⋯-\frac{1}{z^2} - \frac{1}{z^3} - \cdots. If ff is holomorphic on the whole disc ∣z−c∣<R|z - c| < R, then an=0a_n = 0 for n<0n < 0, and the Laurent series is the Taylor series.

Reading path

To understand this page from the ground up, read these 19 pages in order. The foundations come first.

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Complex numbers, object, no evidence levelComplex numbersobjectComplex plane, concept, no evidence levelComplex planeconceptReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionComplex exponential, definition, no evidence levelComplex exponentialdefinitionContour integral, definition, no evidence levelContour integraldefinitionPower series in the complex plane, definition, no evidence levelPower series in the c…definitionReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionSupremum, definition, no evidence levelSupremumdefinitionCircle integral, definition, no evidence levelCircle integraldefinitionLaurent series, definition, no evidence levelLaurent seriesdefinitionCompleteness axiom, statement, no evidence levelCompleteness axiomstatementExistence of square roots of nonnegative reals, statement, no evidence levelExistence of square r…statementSquare root, definition, no evidence levelSquare rootdefinitionModulus of a complex number, definition, no evidence levelModulus of a complex …definitionComplex differentiability, definition, no evidence levelComplex differentiabi…definitionAnnulus, definition, no evidence levelAnnulusdefinitionHolomorphic function, definition, no evidence levelHolomorphic functiondefinitionExistence of Laurent expansions on an annulus, statement, no evidence levelExistence of Laurent …statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Complex numbers object
  2. Complex plane concept
  3. Real numbers object
  4. Set bounded above definition
  5. Complex exponential definition
  6. Contour integral definition
  7. Power series in the complex plane definition
  8. Real and imaginary parts definition
  9. Supremum definition
  10. Circle integral definition
  11. Laurent series definition
  12. Completeness axiom statement
  13. Existence of square roots of nonnegative reals statement
  14. Square root definition
  15. Modulus of a complex number definition
  16. Complex differentiability definition
  17. Annulus definition
  18. Holomorphic function definition
  19. Existence of Laurent expansions on an annulus statement

Where this leads

8 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Laurent_series, Definition
  2. Web https://en.wikipedia.org/wiki/Laurent_series, Uniqueness
About this page
Tag06WHRARZKY
URIm2://06WHRARZKY
Aliaseslaurents-theorem, laurent-expansion-theorem
Kindstatement, theorem
WritingW0 stub
MSC30B10
StatusThis copy of the site has no build data. Run m2 status 06WHRARZKY.
UsesAnnulus, Laurent series, Holomorphic function, Circle integral

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