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Modular Equations and Approximations to π

  • Lennart Berggren
  • Jonathan Borwein
  • Peter Borwein

Abstract

If we suppose that
$$(1 + {e^{ - \pi Nn}})(1 + {e^{ - 3\pi Nn}})(1 + {e^{ - b\pi Nn}}) \ldots = {2^{\frac{1}{4}}}{c^{ - \pi Nn/24}}{G_n} \ldots \ldots \ldots $$
(1)
and
$$(1 - {e^{ - \pi Nn}})(1 - {e^{ - 3\pi Nn}})(1 - {e^{ - b\pi Nn}}) \ldots = {2^{\frac{1}{4}}}{e^{ - \pi Nn/24}}{g_n}, \ldots \ldots \ldots $$
(2)
then G n and g n can always be expressed as roots of algebraical equations when n is any rational number. For we know that
$$(1 + q)(1 + {q^3})(1 + {q^5}) \ldots = {2^{\frac{1}{6}}}{q^{\frac{1}{{24}}}}{(kk')^{ - \frac{1}{{12}}}} \ldots \ldots \ldots \ldots \ldots $$
(3)
and
$$(1 - q)(1 - {q^3})(1 - {q^5}) \ldots = {2^{\frac{1}{6}}}{q^{\frac{1}{{24}}}}{k^{ - \frac{1}{{12}}}}{k'^{\frac{1}{6}}}. \ldots \ldots \ldots \ldots \ldots $$
(4)

Keywords

Rational Number Arithmetical Progression Algebraic Function Modular Equation Finite Term 
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 Science+Business Media New York 2000

Authors and Affiliations

  • Lennart Berggren
    • 1
  • Jonathan Borwein
    • 1
  • Peter Borwein
    • 1
  1. 1.Department of Mathematics and StatisticsSimon Fraser UniversityBurnabyCanada

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