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, Jiirgen Elstrodt nominates as a first candidate the partial fraction expansion of the cotangent function
$$ \pi \cot \pi x = \frac{1}{x} + \sum\limits_{n = 1}^\infty {\left( {\frac{1}{{x + n}} + \frac{1}{{x - n}}} \right)\,(x \in } \backslash )$$
.

Keywords

Sine 

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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 Weierstrafische stetige, nirgends differenzierbare Funktion, Math. Semester-berichte 45 (1998), 207–220.MathSciNetMATHGoogle 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 2004

Authors and Affiliations

  • Martin Aigner
    • 1
  • Günter M. Ziegler
    • 2
  1. 1.Institut für Mathematik II (WE2)Freie Universität BerlinBerlinGermany
  2. 2.Institut für Mathematik, MA 6-2Technische Universität BerlinBerlinGermany

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