Star of a function element

Mittag-Leffler star

A star-like domain in which the given element

of an analytic function (cf. Analytic function, element of an) can be continued analytically along rays issuing from the centre . The star consists of those points of the complex -plane which can be reached by analytic continuation of as a power series along all possible rays from the centre of the series. If , , is a ray on which there are points that cannot be reached this way, then there is a point on the ray such that the element can be continued to any point of the interval but not beyond. If continuation is possible to any point of the ray, one puts . The set of points belonging to all intervals is a (simply-connected) star-like domain about , called the star of the function element and denoted by . Analytic continuation in results in a regular analytic function , which is the univalent branch in of the complete analytic function generated by the given element.

All points of the boundary are accessible (cf. Attainable boundary point). In questions of analytic continuation (see also Hadamard theorem) one also defines angular, attainable and well-attainable points of . A point is called an angular boundary point of the star of a function element if its modulus is minimal among all points of with the same argument . A point is called an attainable boundary point of the star if there is a half-disc such that is regular everywhere inside and at the points of its diameter other than . The point is said to be well-attainable if there is a sector with apex and angle greater than , such that is regular in the domain for sufficiently small .

G. Mittag-Leffler

showed that a regular function can be expressed in its star as a series of polynomials convergent inside :

(*)

Formula (*) is known as the Mittag-Leffler expansion in a star. The degrees of the polynomials and their coefficients , are independent of the form of and can be evaluated once and for all. This was done by P. Painlevé (see [2], [3]).

References

[1a]	G. Mittag-Leffler, "Sur la répresentation analytique d'une branche uniforme d'une fonction monogène I" Acta Math. , 23 (1899) pp. 43–62
[1b]	G. Mittag-Leffler, "Sur la répresentation analytique d'une branche uniforme d'une fonction monogène II" Acta Math. , 24 (1901) pp. 183–204
[1c]	G. Mittag-Leffler, "Sur la répresentation analytique d'une branche uniforme d'une fonction monogène III" Acta Math. , 24 (1901) pp. 205–244
[1d]	G. Mittag-Leffler, "Sur la répresentation analytique d'une branche uniforme d'une fonction monogène IV" Acta Math. , 26 (1902) pp. 353–393
[1e]	G. Mittag-Leffler, "Sur la répresentation analytique d'une branche uniforme d'une fonction monogène V" Acta Math. , 29 (1905) pp. 101–182
[2]	A.I. Markushevich, "Theory of functions of a complex variable" , 2 , Chelsea (1977) (Translated from Russian)
[3]	E. Borel, "Leçons sur les fonctions de variables réelles et les développements en séries de polynômes" , Gauthier-Villars (1905)