= Canonical product construction for zeros escaping to the disk boundary
{title2=$F(z)=z^m\prod_{n\ge1}E_n(z/a_n)$}
For nonzero prescribed zeros $a_n$ with $|a_n|\to1$, the <Weierstrass elementary factors> of increasing order give the displayed <holomorphic function> on the <unit disc>; $m$ handles finitely many zeros at zero. On any compact disc, eventually $|z/a_n|\le q<1$, and $|\log E_n(z/a_n)|\le q^{n+1}/[(n+1)(1-q)]$. The <infinite product convergence from logarithmic tails> gives a nonvanishing tail. The finite initial factors supply exactly the prescribed zeros and <multiplicities>. No <Blaschke condition> is required unless boundedness is imposed.
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