Hadamard Factorization Theorem

Theorem

Let $f: \C \to \C$ be an entire function of finite order $\omega$.

Let $0$ be a zero of $f$ of multiplicity $m \ge 0$.

Let $\sequence {a_n}$ be the sequence of non-zero zeroes of $f$, repeated according to multiplicity.

Then:

$f$ has finite rank $p \le \omega$

and:

there exists a polynomial $g$ of degree at most $\omega$ such that:

$\ds \map f z = z^m e^{\map g z} \prod_{n \mathop = 1}^\infty E_p \paren {\frac z {a_n} }$

where $E_p$ denotes the $p$th Weierstrass elementary factor.

Proof

By Convergence Exponent is Less Than Order, $f$ has finite exponent of convergence $\tau \le \omega$.

By Relation Between Rank and Exponent of Convergence, $f$ has finite rank $p \leq \omega$.

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Also see

• Weierstrass Factorization Theorem
• Order is Maximum of Exponent of Convergence and Degree
• Definition:Hadamard's Canonical Factorization

Source of Name

This entry was named for Jacques Salomon Hadamard.

Sources

• 1932: A.E. Ingham: The Distribution of Prime Numbers: Chapter III: Further Theory of $\map \zeta s$. Applications: $\S7$: Integral Functions: Theorem $F3$