Numerical analytic continuation

Abstract

Let f be an analytic function on a simply-connected compact continuum E of the complex z-plane. This might be an interval of the real line, where f might be real analytic. How can we calculate good estimates of the analytic continuation of f to other points $$z\in {\mathbb C}$$ z ∈ C ? How can we estimate the locations of real or complex singularities of f? We review both the theory and the practice of some existing methods for these problems and propose that excellent results can be obtained from the computation of rational approximations of f by the AAA algorithm. In the case of analytic functions of two or more variables, the rational approximations are applied along line segments or other analytic arcs.

Exponential Asymptotics Using Numerical Rational Approximation in Linear Differential Equations

“…This limitation is a common feature of rational approximation methods, and was studied in depth for Padé approximation in [23], using a quantity known as the “condenser capacity” in the approximation. More recently, an explanation of the branch curve selection for AAA that built on these ideas was presented in [25].…”

Optimal Approximation of Unique Continuation

“…Another well-known ill-posed problem is analytic continuation, which, similarly to unique continuation, possesses conditional stability under the assumption of an a priori bound. We will not discuss this problem here; for its conditional stability/conditioning and numerical approximations, we refer the reader to [46,47] and the references therein.…”