Math 2.0

Functions of a complex variable Definition

Mittag-Leffler star

Builds on Power seriesRadius of convergenceAnalytic continuationPath

Definition

Let a∈ℂa \in \C, and let f(z)=∑k≥0ck(z−a)kf(z) = \sum_{k \ge 0} c_k (z - a)^k be a power series with positive radius of convergence. For z∈ℂz \in \C, let γz ⁣:[0,1]→ℂ\gamma_z \colon [0, 1] \to \C be the segment γz(t)=a+t(z−a)\gamma_z(t) = a + t(z - a). The Mittag-Leffler star of ff with respect to aa is the set of all z∈ℂz \in \C with the following property. ff has an analytic continuation along the path γz\gamma_z.

Remarks

Example: for f(z)=1/(1−z)=∑kzkf(z) = 1/(1 - z) = \sum_k z^k and a=0a = 0, the point z=iz = i lies in the star, because the segment from 00 to ii avoids 11 and 1/(1−z)1/(1-z) is holomorphic on ℂ∖{1}\C \setminus \set{1}. The star is ℂ∖[1,∞)\C \setminus [1, \infty). Non-example: z=2z = 2 is not in the star, because the segment from 00 to 22 passes through the pole 11, where no continuation exists. The star is open, it contains the disc of convergence, and with each point zz it contains the whole segment from aa to zz. For an entire function the star is all of ℂ\C.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

Show the path as a graph
Analytic continuation, concept, no evidence levelAnalytic continuationconceptGraph, definition, no evidence levelGraphdefinitionSequence, definition, no evidence levelSequencedefinitionWalk, definition, no evidence levelWalkdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionPath, definition, no evidence levelPathdefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionPower series, definition, no evidence levelPower seriesdefinitionRadius of convergence, definition, no evidence levelRadius of convergencedefinitionMittag-Leffler star, definition, no evidence levelMittag-Leffler stardefinition
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. Analytic continuation concept
  2. Graph definition
  3. Sequence definition
  4. Walk definition
  5. Real sequence definition
  6. Path definition
  7. Bounded sequence definition
  8. Series definition
  9. Power series definition
  10. Radius of convergence definition
  11. Mittag-Leffler star definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Mittag-Leffler_star
About this page
Tag7ET5N8BWZ6
URIm2://7ET5N8BWZ6
Kinddefinition
WritingW0 stub
MSC30B40, 30B10
DefinesMittag-Leffler star
UsesPower series, Radius of convergence, Analytic continuation, Path
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (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.