In mathematics, and particularly in the field of complex analysis, the Hadamard factorization theorem asserts that every entire function with finite order can be represented as a product involving its zeroes and an exponential of a polynomial.
The theorem may be viewed as an extension of the fundamental theorem of algebra, which asserts that every polynomial may be factored into linear factors, one for each root.
It is closely related to Weierstrass factorization theorem, which does not restrict to entire functions with finite orders.
Define the Hadamard canonical factors
Entire functions of finite order
have Hadamard's canonical representation:[1]
is the smallest non-negative integer such that the series
is called the genus of the entire function
If the order is a positive integer, then there are two possibilities:
Furthermore, Jensen's inequality implies that its roots are distributed sparsely, with critical exponent
are entire functions of genus
Define the critical exponent of the roots of
In other words, we have an asymptotic bound on the growth behavior of the number of roots of the function:
is an entire function with infinitely many roots, then
Note: These two equalities are purely about the limit behaviors of a real number sequence
is also an entire function with the same order
This is the tricky part and requires splitting into two cases.
, we can split the sum to a finite bulk and an infinite tail:
The bulk term is a finite sum, so it converges uniformly.
It remains to bound the tail term.
As usual in analysis, we fix some small
This does not exactly work, however, due to bad behavior of
Consequently, we need to pepper the complex plane with "forbidden disks", one around each
, we can pick an increasing sequence of radii
[nb 2] that avoids these forbidden disks, then by the same application of Borel–Carathéodory theorem,
As usual in analysis, this infinite sum can be split into two parts: a finite bulk and an infinite tail term, each of which is to be separately handled.
With Hadamard factorization we can prove some special cases of Picard's little theorem.
is entire, nonconstant, and has finite order, then it assumes either the whole complex plane or the plane minus a single point.
is entire, nonconstant, and has finite, non-integer order
, then it assumes the whole complex plane infinitely many times.