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''Mittag-Leffler star''
 
''Mittag-Leffler star''
  
 
A [[Star-like domain|star-like domain]] in which the given element
 
A [[Star-like domain|star-like domain]] in which the given element
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s0872301.png" /></td> </tr></table>
+
$$
 
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s0872302.png" />. The star consists of those points of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s0872303.png" />-plane which can be reached by [[Analytic continuation|analytic continuation]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s0872304.png" /> as a power series along all possible rays from the centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s0872305.png" /> of the series. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s0872306.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s0872307.png" />, is a ray on which there are points that cannot be reached this way, then there is a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s0872308.png" /> on the ray such that the element can be continued to any point of the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s0872309.png" /> but not beyond. If continuation is possible to any point of the ray, one puts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723010.png" />. The set of points belonging to all intervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723011.png" /> is a (simply-connected) star-like domain about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723012.png" />, called the star of the function element and denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723013.png" />. Analytic continuation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723014.png" /> results in a regular analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723015.png" />, which is the univalent branch in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723016.png" /> of the [[Complete analytic function|complete analytic function]] generated by the given element.
+
$$
  
All points of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723017.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723018.png" />. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723019.png" /> is called an angular boundary point of the star of a function element if its modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723020.png" /> is minimal among all points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723021.png" /> with the same argument <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723022.png" />. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723023.png" /> is called an attainable boundary point of the star if there is a half-disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723024.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723025.png" /> is regular everywhere inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723026.png" /> and at the points of its diameter other than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723027.png" />. The point is said to be well-attainable if there is a sector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723028.png" /> with apex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723029.png" /> and angle greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723030.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723031.png" /> is regular in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723032.png" /> for sufficiently small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723033.png" />.
+
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.
  
G. Mittag-Leffler
+
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 $.
  
showed that a regular function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723034.png" /> can be expressed in its star as a series of polynomials convergent inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723035.png" />:
+
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} $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723036.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
f ( z )  = \sum _ { n=0 } ^  \infty  \
 +
\sum _ { \nu = 0 } ^ { {k _ n } } c _  \nu  ^ {(n)}
 +
\frac{f ^ { ( \nu ) } ( a ) }{\nu ! }
 +
( z - a ) ^  \nu  .
 +
$$
  
Formula (*) is known as the Mittag-Leffler expansion in a star. The degrees of the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723037.png" /> and their coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723039.png" /> are independent of the form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087230/s08723040.png" /> and can be evaluated once and for all. This was done by P. Painlevé (see [[#References|[2]]], [[#References|[3]]]).
+
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====
 
====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>
 
<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>

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=12951
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article