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]).