Weierstrass elementary factor ($E_p(w)=(1-w)\exp ({\sum }_{k=1}^p{w}^k/k)$)

The Weierstrass elementary factor has a simple zero at $w=1$ and no other zeros. Its logarithmic series begins at degree $p+1$: $\log E_p(w)=-{\sum }_{\nu \ge p+1}{w}^{\nu }/\nu$ 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.