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    Hadamard Factorization Theorem

    Definition

    Let f be an entire function of finite order λ and {a_j} the zeros of f, listed with multiplicity, then the rank p of f is defined as the least positive integer such that sum_(a_n !=0) ( left bracketing bar a_n right bracketing bar )^(-(p + 1))<∞. Then the canonical Weierstrass product is given by f(z) = e^(g(z)) P(z), and g has degree q<=λ. The genus μ of f is then defined as max(p, q), and the Hadamard factorization theory states that an entire function of finite order λ is also of finite genus μ, and μ<=λ.

    Associated person

    Jacques Hadamard

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