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)