Computational analytic continuation

TL;DR

This paper explores the theoretical and practical aspects of analytic continuation for holomorphic functions, establishing limitations, deriving bounds, and demonstrating a computational algorithm for algebraic curves.

Contribution

It proves the impossibility of finite-value-dependent algorithms, derives a computable step size bound, and provides a practical numerical method for algebraic curves.

Findings

Finite-value-dependent algorithms are generally impossible.

A computable local bound on sampling step size is derived.

A numerical example demonstrates practical analytic continuation.

Abstract

Holomorphic functions are amazing because their values in an ever so small disk in the complex plane completely determine the function values at arbitrary points in their maximum possible domain. The process of extending such a function beyond its initial domain is called analytic continuation. We attempt to make this theoretic result tractable by computers. In the present article, we first prove that any algorithm for analytic continuation can generally not depend on finitely many function values only, without closer inspection of the function itself. We then derive a computable local bound on the step size between sampling points which yields an algorithm for analytic continuation of complex plane algebraic curves. Finally, we provide a numerical example demonstrating its practical use.