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Hadamard Factorization Theorem

المؤلف:  Krantz, S. G.

المصدر:  "The Hadamard Factorization Theorem." §9.3.5 in Handbook of Complex Variables. Boston, MA: Birkhäuser

الجزء والصفحة:  pp. 121-122

22-11-2018

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Hadamard Factorization Theorem

Let  be an entire function of finite order  and {a_j}" src="http://mathworld.wolfram.com/images/equations/HadamardFactorizationTheorem/Inline3.gif" style="height:20px; width:22px" /> 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)

 

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REFERENCES:

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