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
- The Mittag-Leffler star of the complex exponential function defined in a neighborhood of a = 0 is the entire complex plane.
- The Mittag-Leffler star of the complex logarithm defined in the neighborhood of point a = 1 is the entire complex plane without the origin and the negative real axis. In general, given the complex logarithm defined in the neighborhood of a point a ≠ 0 in the complex plane, this function can be extended all the way to infinity on any ray starting at a, except on the ray which goes from a to the origin, one cannot extend the complex logarithm beyond the origin along that ray.
- Any open star-convex set is the Mittag-Leffler star of some complex-analytic function, since any open set in the complex plane is a domain of holomorphy.
Uses
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
- Shenitzer, Abe; Stillwell, John, eds. (2002). Mathematical evolutions. Washington, DC: Mathematical Association of America. p. 32. ISBN 0-88385-536-4.
- Korevaar, Jacob (2004). Tauberian theory: a century of developments. Berlin; New York: Springer. ISBN 3-540-21058-X.
External links
 | This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (September 2015) (Learn how and when to remove this message) |
- E.D. Solomentsev (2001) , "Star_of_a_function_element", Encyclopedia of Mathematics, EMS Press