= Prescribed zeros in a plane domain via Runge approximation
Every <closed discrete subset> of a plane <domain> is the zero set of a <holomorphic function>, with assigned finite positive multiplicities if desired. Exhaust the domain by relatively filled compact sets. For a prescribed point outside an earlier compact, choose a rational factor with that zero and its possible pole outside the domain in the same complementary component. A slit joining zero and pole allows a <holomorphic logarithm> near the earlier compact. Approximate that logarithm by <Runge theorem> and multiply the rational factor by the negative exponential of the approximant. The resulting factors tend to one with summable errors on each compact. Their <infinite product> has the prescribed zeros and no additional zeros.
// Destination: analysis.bigb
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