Cotangent and the Herglotz trick

  • Martin Aigner
  • Günter M. Ziegler

Abstract

What is the most interesting formula involving elementary functions? In his beautiful article [2], whose exposition we closely follow, Jürgen Elstrodt nominates as a first candidate the partial fraction expansion of the cotangent function:
$$ \pi \cot \pi x = \frac{1}{x} + \sum^{\infty}_{n=1} (\frac{1}{x+n} + \frac{1}{x-n}) \quad (x \in \mathbb{R}\textbackslash \mathbb{Z}). $$

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References

  1. [1]
    S. Bochner: Book review of “Gesammelte Schriften” by Gustav Herglotz, Bulletin Amer. Math. Soc. 1 (1979), 1020-1022.MathSciNetCrossRefGoogle Scholar
  2. [2]
    J. Elstrodt: Partialbruchzerlegung des Kotangens, Herglotz-Trick und die Weierstraßsche stetige, nirgends differenzierbare Funktion, Math. Semesterberichte 45 (1998), 207-220.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    L. Euler: Introductio in Analysin Infinitorum, Tomus Primus, Lausanne 1748; Opera Omnia, Ser. 1, Vol. 8. In English: Introduction to Analysis of the Infinite, Book I (translated by J. D. Blanton), Springer-Verlag, New York 1988.Google Scholar
  4. [4]
    L. Euler: Institutiones calculi differentialis cum ejus usu in analysi finitorum ac doctrina serierum, Petersburg 1755; Opera Omnia, Ser. 1, Vol. 10.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Martin Aigner
    • 1
  • Günter M. Ziegler
    • 1
  1. 1.FU BerlinBerlinGermany

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