Laplace measure transitions and ghosts for meromorphic functions

TL;DR

This paper investigates measure transitions for Laplace transforms of meromorphic functions, deriving explicit formulas, and constructs a counterexample demonstrating non-uniqueness in the heat equation Cauchy problem.

Contribution

It provides explicit transition formulas for measures associated with meromorphic Laplace transforms and constructs a counterexample to solution uniqueness in the heat equation.

Findings

Measures coincide when no poles are on the separatrix.

Explicit transition formulas for finitely many poles.

Counterexample to heat equation solution uniqueness.

Abstract

We study the measure transition problem for bilateral Laplace transforms of meromorphic functions on vertical strips. Given a meromorphic function F admitting Laplace representations on two adjacent strips separated by a vertical line, we investigate how the corresponding determining measures are related. Our first result shows that in the absence of poles on the separatrix the determining measures coincide. We next derive explicit transition formulas for the case of finitely many poles and obtain sufficient conditions under which these formulas remain valid for infinitely many poles. Applications are given to the analytic continuation of the zeta function, periodic and almost periodic functions, and quotients of Gamma functions related to the confluent hypergeometric function. Finally, using generalized Cauchy integrals, we construct an entire function admitting distinct Laplace representations on the right and left half-planes, thereby producing a ghost transition. This provides a counterexample to uniqueness of solutions of the Cauchy problem for the heat equation.