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. It is named for Jacques Hadamard.

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.

Formal statement

Define the Hadamard canonical factorsEntire functions of finite order have Hadamard's canonical representation: are those roots is the smallest non-negative integer.

.

Convergence exponent

Define the convergence exponent as the following:.

Theorem:Note: These two equalities are purely about the limit behaviors of a real number that diverges to infinity. It does not involve complex analysis.

Proposition:, by Jensen's formula.

Applications

With Hadamard factorization we can prove some special cases of Picard's little theorem.

Theorem: is entire, nonconstant, and has finite order, then it assumes either the whole complex plane or the plane minus a single point.

Proof:. By the fundamental theorem of algebra assumes all nonzero values.

Theorem:, then it assumes the whole complex plane infinitely many times.

Proof: is an integer.

Proof

The proof below follows Conway's treatment of Hadamard's factorization theorem.

at the origin, write

can then be restored at the end.

, repeated according to multiplicity and ordered so that

, and in particular that

, counted with multiplicity. By Jensen's formula, one obtains an estimate of the form

where

,

, it follows that

, this gives

. Therefore

. Hence

Consequently the canonical product

, with the same multiplicities. Thus

such that

Hence

. For this Conway uses the following logarithmic-derivative lemma.

Lemma.,

gives

,

For a single elementary factor,

times, the polynomial part vanishes, and hence

Comparing the two formulas gives

.

Restoring the zero at the origin, one obtains

. This is Hadamard's factorization theorem.

References