An asymptotic expansion for the semi‐infinite sum of Dirac‐δ functions |
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| Authors: | Otto Rendón Leonardo Di G Sigalotti Jaime Klapp |
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| Affiliation: | 1. Centro de FísicaInstituto Venezolano de Investigaciones Científicas (IVIC);2. Departamento de Física‐FACYT, Universidad de Carabobo, Valencia, Edo. Carabobo, Venezuela;3. área de Física de Procesos Irreversibles, Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana—Azcapotzalco (UAM‐A), 02200 Mexico City, Mexico;4. Departamento de FísicaInstituto Nacional de Investigaciones Nucleares (ININ), Carretera México‐Toluca km. 36.5, La Marquesa, 52750 Ocoyoacac, Estado de México, Mexico;5. ABACUS—Centro de Matemáticas Aplicadas y Cómputo de Alto Rendimiento, Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados (Cinvestav‐IPN), Ocoyoacac, Estado de México, Mexico |
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| Abstract: | In this paper, we derive an asymptotic expansion for the semi‐infinite sum of Dirac‐δ functions centered at discrete equidistant points defined by the set  . The method relies on the Laplace transform of the semi‐infinite sum of Dirac‐δ functions. The derived series distribution takes the form of the Euler‐Maclaurin summation when the distributions are defined for complex or real‐valued continuous functions over the interval  . For n=1, the series expansion contributes with a term equal to δ(x)/2, which survives in the limit when a→0+. This term represents a correction term, which is in general omitted in calculations of the density of states of quantum confined systems by finite‐size effects. |
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| Keywords: | Bernoulli and Euler numbers and polynomials distributions distribution spaces generalized functions real functions series expansions set theory |
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. The method relies on the Laplace transform of the semi‐infinite sum of Dirac‐δ functions. The derived series distribution takes the form of the Euler‐Maclaurin summation when the distributions are defined for complex or real‐valued continuous functions over the interval 
. For n=1, the series expansion contributes with a term equal to δ(x)/2, which survives in the limit when a→0+. This term represents a correction term, which is in general omitted in calculations of the density of states of quantum confined systems by finite‐size effects.