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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:
$$ \pi \cot \pi x = \frac{1}{x} + \sum^{\infty}_{n=1} (\frac{1}{x+n} + \frac{1}{x-n}) \quad (x \in \mathbb{R}\textbackslash \mathbb{Z}). $$

Keywords

Functional Equation Zeta Function Elementary Function Power Series Expansion Riemann Zeta Function 
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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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.MathSciNetzbMATHCrossRefGoogle 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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