View source for Genus of an entire function

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The integer equal to the larger of the two numbers  $  p $
and  $  q $
in the representation of the [[Entire function|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====
<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====
The genus plays a role in factorization theorems for entire functions, cf. e.g. [[Hadamard theorem|Hadamard theorem]]; [[Weierstrass theorem|Weierstrass theorem]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.P. Boas,   "Entire functions" , Acad. Press  (1954)</TD></TR></table>