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''Mittag-Leffler star''

A [[Star-like domain|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|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|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|complete analytic function]] generated by the given element.

All points of the boundary  $  \partial  S _ {f} $
are accessible (cf. [[Attainable boundary point|Attainable boundary point]]). In questions of analytic continuation (see also [[Hadamard theorem|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 [[#References|[2]]], [[#References|[3]]]).

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

{{SourceattributionEOM|Star of a function element}}
[[Category:Mathematics]]