Mittag-Leffler star

Illustration of the Mittag-Leffler star (the region bounded by the blue contour). The original disk U is centered at a.

In complex analysis, a branch of mathematics, the Mittag-Leffler star of a complex-analytic function is a set in the complex plane obtained by attempting to extend that function along rays emanating from a given point. This concept is named after Gösta Mittag-Leffler.

Definition and elementary properties

Formally, the Mittag-Leffler star of a complex-analytic function ƒ defined on an open disk U in the complex plane centered at a point a is the set of all points z in the complex plane such that ƒ can be continued analytically along the line segment joining a and z (see analytic continuation along a curve).

It follows from the definition that the Mittag-Leffler star is an open star-convex set (with respect to the point a) and that it contains the disk U. Moreover, ƒ admits a single-valued analytic continuation to the Mittag-Leffler star.

Examples

Uses

An illustration of the regions of convergence of the Mittag-Leffler expansion and the Taylor series expansion around a (the regions bounded by the blue curve and red circle respectively).

Any complex-analytic function ƒ defined around a point a in the complex plane can be expanded in a series of polynomials which is convergent in the entire Mittag-Leffler star of ƒ at a. Each polynomial in this series is a linear combination of the first several terms in the Taylor series expansion of ƒ around a.

Such a series expansion of ƒ, called the Mittag-Leffler expansion, is convergent in a larger set than the Taylor series expansion of ƒ at  a. Indeed, the largest open set on which the latter series is convergent is a disk centered at a and contained within the Mittag-Leffler star of ƒ at a

References

This article is issued from Wikipedia - version of the 6/2/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.