Difference between revisions of "Genus of an entire function"
From Encyclopedia of Mathematics
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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 | ||
| − | The number | + | $$ |
| + | \sum_{k=1} ^ \infty | ||
| + | |||
| + | \frac{1}{| a _ {k} | ^ {p + 1 } } | ||
| + | |||
| + | < \infty . | ||
| + | $$ | ||
| + | |||
| + | The number $ p $ | ||
| + | is called the genus of the product appearing in formula (*). | ||
====References==== | ====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> | ||
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====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
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