Bayesian parametric analytic continuation of Green's functions

Analytic continuation of multiple Hurwitz zeta functions

We use a variant of a method of Goncharov, Kontsevich, and Zhao [Go2, Z] to meromorphically continue the multiple Hurwitz zeta function ζd(s; θ) = ∑ 0<n1<···<nd (n1 + θ1) −s1 · · · (nd + θd)d , θk ∈ [0, 1), to C, to locate the hyperplanes containing its possible poles, and to compute the residues at the poles. We explain how to use the residues to locate trivial zeros of ζd(s; θ).

Analytic Continuation of Multiple Zeta Functions

In this paper we shall define the analytic continuation of the multiple (Euler-Riemann-Zagier) zeta functions of depth d: ζ(s1, . . . , sd) := ∑ 0 1 and ∑d j=1 Re (sj) > d. We shall also study their behavior near the poles and pose some open problems concerning their zeros and functional equations at the end.

Numerical analytic continuation of holomorphic functions in Cn

Analytic Continuation of Massless Two-Loop Four-Point Functions

We describe the analytic continuation of two-loop four-point functions with one off-shell external leg and internal massless propagators from the Euclidean region of space-like 1→ 3 decay to Minkowskian regions relevant to all 1→ 3 and 2→ 2 reactions with one space-like or time-like off-shell external leg. Our results can be used to derive two-loop master integrals and unrenormalized matrix ele...

On Effective Analytic Continuation

Until now, the area of symbolic computation has mainly focused on the manipulation of algebraic expressions. It would be interesting to apply a similar spirit of “exact computations” to the field of mathematical analysis. One important step for such a project is the ability to compute with computable complex numbers and computable analytic functions. Such computations include effective analytic...