Weierstrass elementary factor
= Weierstrass elementary factor
{c}
{title2=$E_p(w)=(1-w)\exp(\sum_{k=1}^p w^k/k)$}
The Weierstrass elementary factor has a simple zero at $w=1$ and no other zeros. Its logarithmic series begins at degree $p+1$: $\log E_p(w)=-\sum_{\nu\geq p+1}w^\nu/\nu$ for $|w|<1$. Raising the factor order $p$ makes the tail arbitrarily small on a fixed compact set and enables <infinite product convergence from logarithmic tails> for unrestricted discrete zero data.