Optimized Polynomial Expansion for Scattering Amplitudes

Abstract

In devising the most efficient way to determine a scattering amplitude from experimental data, it is important to make full use of the analyticity properties of the amplitude. The amplitude $f\left(x\right)\mathrm{}$ considered here is given by data on a connected part of the real $x$ axis, and $f\left(x\right)\mathrm{}$ is assumed to be analytic in a simply connected part of the $x$ plane; there are branch cuts on part of the remainder of the real axis. For convenience in practical calculations it is simplest to expand in polynomials, but for greater flexibility one may consider polynomials of some function $z\left(x\right)\mathrm{}$. The polynomial expansion will converge as rapidly as possible if $z\left(x\right)\mathrm{}$ maps the domain of analyticity in the $x$ plane onto the interior of a certain ellipse in the $z$ plane. More precisely, the expansion will then have the greatest possible geometric rates of convergence, both to $f\left(x\right)\mathrm{}$ in the physical region, and also at any arbitrary point away from the physical region to which one may wish to extrapolate. Formulas are given that enable the mapping from a cut plane to an ellipse to be calculated quickly and easily. Some properties of the transformation that are relevant to partial-wave analysis are examined in detail. A method is suggested whereby the requirements of unitarity may be explicitly incorporated.