Star of a function element

Mittag-Leffler star

A star-like domain in which the given element

$$ f ( z ) = \sum _ { k=0 } ^ \infty c _ {k} ( z - a ) ^ {k} $$

of an analytic function (cf. Analytic function, element of an) can be continued analytically along rays issuing from the centre $ a $. The star consists of those points of the complex $ z $-plane which can be reached by analytic continuation of $ f ( z ) $ as a power series along all possible rays from the centre $ a $ of the series. If $ z = a + r e ^ {i \phi } $, $ 0 \leq r < + \infty $, is a ray on which there are points that cannot be reached this way, then there is a point $ z _ {1} \neq a $ on the ray such that the element can be continued to any point of the interval $ [ a , z _ {1} ) $ but not beyond. If continuation is possible to any point of the ray, one puts $ z _ {1} = \infty $. The set of points belonging to all intervals $ [ a , z _ {1} ) $ is a (simply-connected) star-like domain about $ a $, called the star of the function element and denoted by $ S _ {f} $. Analytic continuation in $ S _ {f} $ results in a regular analytic function $ f ( z ) $, which is the univalent branch in $ S _ {f} $ of the complete analytic function generated by the given element.

All points of the boundary $ \partial S _ {f} $ 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 $ \partial S _ {f} $. A point $ z _ {1} \in \partial S _ {f} $ is called an angular boundary point of the star of a function element if its modulus $ | z _ {1} | $ is minimal among all points of $ \partial S _ {f} $ with the same argument $ \mathop{\rm arg} z _ {1} $. A point $ z _ {1} \in \partial S _ {f} $ is called an attainable boundary point of the star if there is a half-disc $ V ( z _ {1} ) $ such that $ f ( z ) $ is regular everywhere inside $ V ( z _ {1} ) $ and at the points of its diameter other than $ z _ {1} $. The point is said to be well-attainable if there is a sector $ V ( z _ {1} ) $ with apex $ z _ {1} $ and angle greater than $ \pi $, such that $ f ( z ) $ is regular in the domain $ \{ V ( z _ {1} ) \cap ( | z - z _ {1} | < \delta ) \} $ for sufficiently small $ \delta > 0 $.

G. Mittag-Leffler showed that a regular function $ f ( z ) $ can be expressed in its star as a series of polynomials convergent inside $ S _ {f} $:

$$ \tag{* } f ( z ) = \sum _ { n=0 } ^ \infty \ \sum _ { \nu = 0 } ^ { {k _ n } } c _ \nu ^ {(n)} \frac{f ^ { ( \nu ) } ( a ) }{\nu ! } ( z - a ) ^ \nu . $$

Formula (*) is known as the Mittag-Leffler expansion in a star. The degrees of the polynomials $ k _ {n} $ and their coefficients $ c _ {0} ^ {(n)} \dots c _ {k _ {n} } ^ {(n)} $, $ n = 0 , 1 \dots $ are independent of the form of $ f ( z ) $ 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)