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The Poisson Sum Formula and Applications

  • Hans Rademacher
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 169)

Abstract

Let f(x) be continuous for all x. We shall presently subject this function to some further restrictions. The problem is to evaluate
$$S = \sum\limits_{n = - \infty }^\infty {f(n).} $$
Poisson’s basic idea is to introduce a variable and to consider
$$S(u) = \sum\limits_{n = - \infty }^\infty {f(n + u)} $$
(35.1)
under suitable conditions of convergence. It is clear that
$$S(u + 1) = \,S(u),$$
(35.11)
since the effect of an increase of u by 1 on the right-hand member of (35.1) is the same as a change of n into n + 1, which leaves the set of integers n(— ∞ < n < ∞) unchanged.

Keywords

Real Axis Uniform Convergence Transformation Formula Slight Generalization Analytic Number Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag, Berlin • Heidelberg 1973

Authors and Affiliations

  • Hans Rademacher

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