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Laplace measure transitions and ghosts for meromorphic functions

João Fontinha, Jorge Buescu, Jaouen Ramalho

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.

Laplace measure transitions and ghosts for meromorphic functions

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.

Paper Structure

This paper contains 7 sections, 13 theorems, 86 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation and definitions
  3. The transition formula
  4. The case of infinitely many poles
  5. Examples and applications
  6. Ghost transitions
  7. Connection to the heat equation

Key Result

Lemma 2.7

If $\mu(t)$ is a normalized function of bounded variation in every finite interval, and if the Laplace integral converges in the strip $\Omega_{a,b}$, then for all $t$ where $\mu(\pm \infty)$ denote the limits $\lim_{t \to \pm \infty} \mu(t).$$\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure 1: Contour $\Gamma_{r, \delta}$
  • Figure 2: Graphs of the measures $\mu_{0,1}$ and $\mu_{1,\infty}$.
  • Figure 3: Polarity of the quotient $\frac{\Gamma(a-s)}{\Gamma(b-s)}$ when $b-a$ is an integer.
  • Figure 4: Contour $\gamma_M$ and region $D_M$.

Theorems & Definitions (39)

  • Definition 2.1: Normalized determining function
  • Definition 2.2: Laplace pair
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Lemma 2.7: Inversion theorem for the Laplace transform
  • proof
  • Lemma 2.8: Order on vertical lines
  • proof
  • ...and 29 more