Hadamard factorization theorem
= Hadamard factorization theorem
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= Hadamard factorization
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An <entire function> of finite order factors as its canonical zero product times the exponential of a <polynomial>. For order at most one and nonzero value at zero, it has the form $f(z)=f(0)e^{Bz}\prod_j(1-z/a_j)e^{z/a_j}$. Genus-one factors converge when $\sum_j|a_j|^{-2}<\infty$. A zero-free function of this order is therefore the exponential of an affine <polynomial>.