---
tag: C21ECQ1H84
kind: concept
name: Mittag-Leffler star
msc: ["30B40", "30B10"]
sources:
  - url: https://en.wikipedia.org/wiki/Mittag-Leffler_star
same_as:
  wikidata: Q4189397
  wikipedia: Mittag-Leffler star
---

## Summary

The Mittag-Leffler star of a power series is the set of points that can be reached from its centre by [[analytic continuation|0M3YQJ81NB]] along straight segments.
It is an open set that is star-shaped with respect to the centre, and it contains the disc of convergence.
Mittag-Leffler showed that the function is given on its whole star by a series of polynomials.
The star is a natural domain for representing a function by summation methods beyond the disc of convergence.

## Explanations

### L1 human

Start at the centre of a power series and walk outwards along a ray.
Continue the function along the ray as long as possible, and stop at the first singular point.
The points reached on all rays form the star.
For $1/(1 - z)$ with centre $0$, every ray except the positive real axis can be followed forever, and the positive real axis is blocked at $1$.
So the star is the plane with the half-line $[1, \infty)$ removed.