Canonical product

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Weierstrass canonical product

An entire function with a given sequence of complex numbers as its zeros. Suppose that the zeros are arranged in monotone increasing order of their moduli, , and have no limit point in the finite plane (a necessary condition), i.e. . Then the canonical product has the form


The are called the elementary factors of Weierstrass. The exponents are chosen so that the canonical product is absolutely and uniformly convergent on any compact set; for example, it suffices to take . If the sequence has a finite exponent of convergence

then all the can be chosen to be the same, starting, e.g. from the minimal requirement that ; this is called the genus of the canonical product. If , i.e. if diverges for any , then one has a canonical product of infinite genus. The order of a canonical product (for the definition of the type of a canonical product, see [1]).


[1] B.Ya. Levin, "The distribution of zeros of entire functions" , Amer. Math. Soc. (1980) (Translated from Russian)


See also Blaschke product; Entire function; Hadamard theorem.

How to Cite This Entry:
Canonical product. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article