Pade Approximants of the Green Function in Spherically Symmetric Spacetimes

TL;DR

This paper develops Pade approximants for the scalar Green function in spherically symmetric spacetimes, extending the series expansion beyond its convergence radius to improve understanding of the Green function's behavior near the boundary.

Contribution

It introduces a method to extend the series expansion of the Green function using Pade approximants, providing better convergence properties near the boundary.

Findings

Pade approximants extend series convergence beyond its original radius.

High-order series expansion of the function V(x,x') was achieved.

The method improves the analysis of Green functions near the boundary.

Abstract

We investigate the scalar Green function for spherically symmetric spacetimes expressed as a coordinate series expansion in the separation of the points. We calculate the series expansion of the function $V(x,x')$ appearing in the Hadamard parametrix of the scalar Green function to very high order. This expansion is then used to investigate the convergence properties of the series and to estimate its radius of convergence. Using the method of Pade approximants, we show that the series can be extended beyond its radius of convergence to within a short distance of the normal neighborhood boundary.