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Difference between revisions of "Genus of an entire function"

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The integer equal to the larger of the two numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044030/g0440301.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044030/g0440302.png" /> in the representation of the [[Entire function|entire function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044030/g0440303.png" /> in the form
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<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/g/g044/g044030/g0440304.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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<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/g/g044/g044030/g0440305.png" /></td> </tr></table>
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The integer equal to the larger of the two numbers  $  p $
 +
and  $  q $
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in the representation of the [[Entire function|entire function]]  $  f ( z) $
 +
in the form
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044030/g0440306.png" /> is the degree of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044030/g0440307.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044030/g0440308.png" /> is the least integer satisfying the condition
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$$ \tag{* }
 +
f ( z) = z  ^  \lambda  e ^ {Q ( z) } \prod_{k=1} ^  \infty 
 +
\left ( 1 -
 +
\frac{z}{a _ {k} }
 +
\right )  \mathop{\rm exp}  \left (
 +
\frac{z}{a _ {k} }
 +
+
 +
\frac{z  ^ {2} }{2a _ {k}  ^ {2} }
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+ {} \dots +
 +
\frac{z  ^ {p} }{pa _ {k}  ^ {p} }
 +
\right ) ,
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$$
  
<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/g/g044/g044030/g0440309.png" /></td> </tr></table>
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where  $  q $
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is the degree of the polynomial  $  Q ( z) $
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and  $  p $
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is the least integer satisfying the condition
  
The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044030/g04403010.png" /> is called the genus of the product appearing in formula (*).
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$$
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\sum_{k=1} ^  \infty 
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 +
\frac{1}{| a _ {k} | ^ {p + 1 } }
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 +
<  \infty .
 +
$$
 +
 
 +
The number $  p $
 +
is called the genus of the product appearing in formula (*).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.Ya. Levin,  "Distribution of zeros of entire functions" , Amer. Math. Soc.  (1964)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.Ya. Levin,  "Distribution of zeros of entire functions" , Amer. Math. Soc.  (1964)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 12:51, 6 January 2024


The integer equal to the larger of the two numbers $ p $ and $ q $ in the representation of the entire function $ f ( z) $ in the form

$$ \tag{* } f ( z) = z ^ \lambda e ^ {Q ( z) } \prod_{k=1} ^ \infty \left ( 1 - \frac{z}{a _ {k} } \right ) \mathop{\rm exp} \left ( \frac{z}{a _ {k} } + \frac{z ^ {2} }{2a _ {k} ^ {2} } + {} \dots + \frac{z ^ {p} }{pa _ {k} ^ {p} } \right ) , $$

where $ q $ is the degree of the polynomial $ Q ( z) $ and $ p $ is the least integer satisfying the condition

$$ \sum_{k=1} ^ \infty \frac{1}{| a _ {k} | ^ {p + 1 } } < \infty . $$

The number $ p $ is called the genus of the product appearing in formula (*).

References

[1] B.Ya. Levin, "Distribution of zeros of entire functions" , Amer. Math. Soc. (1964) (Translated from Russian)

Comments

The genus plays a role in factorization theorems for entire functions, cf. e.g. Hadamard theorem; Weierstrass theorem.

References

[a1] R.P. Boas, "Entire functions" , Acad. Press (1954)
How to Cite This Entry:
Genus of an entire function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Genus_of_an_entire_function&oldid=18397
This article was adapted from an original article by A.F. Leont'ev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article