# Hadamard Factorization Theorem

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## Theorem

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

Let $0$ be a zero of $f$ of multiplicity $m\geq0$.

Let $\left\langle{a_n}\right\rangle$ be the sequence of nonzero zeroes of $f$, repeated according to multiplicity.

Then $f$ has finite rank $p\leq\omega$ and there exists a polynomial $g$ of degree at most $\omega$ such that:

- $\displaystyle f \left({z}\right) = z^m e^{g(z)} \prod_{n \mathop = 1}^\infty E_p\left( \frac z{a_n} \right)$

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

## Proof

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

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

## 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 $\zeta(s)$. Applications: $\S7$: Integral Functions: Theorem $F3$