An acceleration method for the power series of entire functions of order $1$

Abstract:

When $f(z)$ is given by a known power series expansion, it is possible to construct the power series expansion for $f(z;p) = {e^{ - pz}}f(z)$. We define ${p_{{\text {opt}}}}$ to be the value of p for which the expansion for $f(z;p)$ converges most rapidly. When $f(z)$ is an entire function of order 1, we show that ${p_{{\text {opt}}}}$ is uniquely defined and may be characterized in terms of the set of singularities ${z_i} = 1/{\sigma _i}$ of an associated function $h(z)$. Specifically, it is the center of the smallest circle in the complex plane which contains all points ${\sigma _i}$.