Math 2.0

Functions of a complex variable Definition

Principal part

Builds on Laurent seriesExistence of Laurent expansions on an annulus

Definition

Let c∈ℂc \in \mathbb{C}, let R∈(0,∞]R \in (0, \infty], and let ff be holomorphic on the punctured disc 0<∣z−c∣<R0 < |z - c| < R. Let f(z)=∑n∈ℤan(z−c)nf(z) = \sum_{n \in \mathbb{Z}} a_n (z - c)^n be the Laurent expansion of ff on this punctured disc. The principal part of ff at cc is the series ∑n=1∞a−n(z−c)−n,\sum_{n = 1}^{\infty} a_{-n} (z - c)^{-n}, the part of the Laurent series with the negative powers of z−cz - c.

Remarks

The principal part converges for each z≠cz \neq c. The difference f(z)−∑n≥1a−n(z−c)−nf(z) - \sum_{n \ge 1} a_{-n} (z - c)^{-n} extends to a holomorphic function on ∣z−c∣<R|z - c| < R. The principal part is 00 at a removable singularity, it has finitely many nonzero terms at a pole, and it has infinitely many at an essential singularity. For example, the principal part of ezz2\frac{e^z}{z^2} at 00 is 1z2+1z\frac{1}{z^2} + \frac{1}{z}.

Reading path

To understand this page from the ground up, read these 20 pages in order. The foundations come first.

Show the path as a graph
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 …statementPrincipal part, definition, no evidence levelPrincipal partdefinition
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.
  • no evidence level
  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
  20. Principal part definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Principal_part, Laurent series definition
  2. Web https://en.wikipedia.org/wiki/Laurent_series, Principal part
About this page
Tag0NDV5R6Z6X
URIm2://0NDV5R6Z6X
Kinddefinition
WritingW1 stated
MSC30B10
DefinesPrincipal part
UsesLaurent series, Existence of Laurent expansions on an annulus

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.