Laplace measure transitions and ghosts for meromorphic functions
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911656378892288 |
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| author | Fontinha, João Buescu, Jorge Ramalho, Jaouen |
| author_facet | Fontinha, João Buescu, Jorge Ramalho, Jaouen |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_04162 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Laplace measure transitions and ghosts for meromorphic functions Fontinha, João Buescu, Jorge Ramalho, Jaouen Complex Variables Primary 44A10, Secondary 30D10, 30E20, 35K05, 42A82 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. |
| title | Laplace measure transitions and ghosts for meromorphic functions |
| topic | Complex Variables Primary 44A10, Secondary 30D10, 30E20, 35K05, 42A82 |
| url | https://arxiv.org/abs/2604.04162 |