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An asymptotic expansion for the semi‐infinite sum of Dirac‐ δ functions
Author(s) -
Rendón Otto,
Sigalotti Leonardo Di G.,
Klapp Jaime
Publication year - 2017
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4479
Subject(s) - mathematics , dirac (video compression format) , laplace transform , asymptotic expansion , mathematical analysis , term (time) , series (stratigraphy) , limit (mathematics) , euler's formula , infinite set , interval (graph theory) , dirac comb , pure mathematics , set (abstract data type) , mathematical physics , dirac equation , combinatorics , dirac algebra , quantum mechanics , dirac spinor , paleontology , physics , computer science , neutrino , biology , programming language
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 N a = { x ∈ R : ∃ n ∈ N ∧ x = n a , ∀ a > 0 } . 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 [ 0 , ∞ ) . 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.