Namespaces
Variants
Actions

Difference between revisions of "Genus of an entire function"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (tex encoded by computer)
(latex details)
 
Line 17: Line 17:
  
 
$$ \tag{* }
 
$$ \tag{* }
f ( z) =
+
f ( z) = z  ^  \lambda  e ^ {Q ( z) } \prod_{k=1} ^  \infty   
$$
 
 
 
$$
 
= \
 
z  ^  \lambda  e ^ {Q ( z) } \prod _ { k= } 1 ^  \infty   
 
 
\left ( 1 -  
 
\left ( 1 -  
 
\frac{z}{a _ {k} }
 
\frac{z}{a _ {k} }
Line 40: Line 35:
  
 
$$  
 
$$  
\sum _ { k= } 1 ^  \infty   
+
\sum_{k=1} ^  \infty   
  
 
\frac{1}{| a _ {k} | ^ {p + 1 } }
 
\frac{1}{| a _ {k} | ^ {p + 1 } }

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=54877
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