Canonical product

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

where

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]).

References

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

Comments

See also Blaschke product; Entire function; Hadamard theorem.