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Difference between revisions of "Star of a function element"

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$$  
 
$$  
f ( z )  =  \sum _ { k= } 0 ^  \infty  c _ {k} ( z - a )  ^ {k}
+
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 $.  
 
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 $-
+
The star consists of those points of the complex  $  z $-plane which can be reached by [[Analytic continuation|analytic continuation]] of  $  f ( 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 $
 
as a power series along all possible rays from the centre  $  a $
 
of the series. If  $  z = a + r e ^ {i \phi } $,  
 
of the series. If  $  z = a + r e ^ {i \phi } $,  
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for sufficiently small  $  \delta > 0 $.
 
for sufficiently small  $  \delta > 0 $.
  
G. Mittag-Leffler
+
G. Mittag-Leffler showed that a regular function  $  f ( z ) $
 
 
showed that a regular function  $  f ( z ) $
 
 
can be expressed in its star as a series of polynomials convergent inside  $  S _ {f} $:
 
can be expressed in its star as a series of polynomials convergent inside  $  S _ {f} $:
  
 
$$ \tag{* }
 
$$ \tag{* }
f ( z )  =  \sum _ { n= } 0 ^  \infty  \  
+
f ( z )  =  \sum _ { n=0 } ^  \infty  \  
\sum _ { \nu = 0 } ^ { {k _ n } } c _  \nu  ^ {(} n)  
+
\sum _ { \nu = 0 } ^ { {k _ n } } c _  \nu  ^ {(n)}
 
\frac{f ^ { ( \nu ) } ( a ) }{\nu ! }
 
\frac{f ^ { ( \nu ) } ( a ) }{\nu ! }
 
  ( z - a )  ^  \nu  .
 
  ( z - a )  ^  \nu  .
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Formula (*) is known as the Mittag-Leffler expansion in a star. The degrees of the polynomials  $  k _ {n} $
 
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) $,  
+
and their coefficients  $  c _ {0}  ^ {(n)} \dots c _ {k _ {n}  }  ^ {(n)} $,  
 
$  n = 0 , 1 \dots $
 
$  n = 0 , 1 \dots $
 
are independent of the form of  $  f ( z ) $
 
are independent of the form of  $  f ( z ) $

Latest revision as of 19:11, 19 June 2020


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)
How to Cite This Entry:
Star of a function element. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Star_of_a_function_element&oldid=49787
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article