The Poisson Sum Formula and Applications

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


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
$$S(u) = \sum\limits_{n = - \infty }^\infty {f(n + u)} $$
under suitable conditions of convergence. It is clear that
$$S(u + 1) = \,S(u),$$
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.




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© Springer-Verlag, Berlin • Heidelberg 1973

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  • Hans Rademacher

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