= Weierstrass factorization theorem
{c}
{title2=$F(z)=z^m e^{h(z)}\prod_j E_{p_j}(z/a_j)^{n_j}$}
A nonzero <entire function> can be described by its discrete zero locations and multiplicities, a product of <Weierstrass elementary factors> and a zero-free exponential factor. Conversely, consistent prescribed multiplicities at distinct points escaping every compact subset can be realized by choosing factor orders large enough for <infinite product convergence from logarithmic tails>. Repeated locations with incompatible exact multiplicities are not admissible data, and a finite accumulation of distinct zeros is excluded by the <identity theorem>.
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