Hadamard Factorization Theorem

Let  be an entire function of finite order  and  the zeros of , listed with multiplicity, then the rank  of  is defined as the least positive integer such that

(1)

Then the canonical Weierstrass product is given by

(2)

and  has degree . The genus  of  is then defined as , and the Hadamard factorization theory states that an entire function of finite order  is also of finite genus , and

(3)

REFERENCES:

Krantz, S. G. "The Hadamard Factorization Theorem." §9.3.5 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 121-122, 1999.