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Integrals evaluated in terms of Catalan's constant

Published online by Cambridge University Press:  03 February 2017

Graham Jameson  and
Nick Lord
Graham Jameson
Affiliation:
Dept. of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF e-mail: [email protected]
Nick Lord
Affiliation:
Tonbridge School, Tonbridge, Kent TN9 1JP e-mail: [email protected]
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Extract

Catalan's constant, named after E. C. Catalan (1814-1894) and usually denoted by G, is defined by

It is, of course, a close relative of

The numerical value is G ≈ 0.9159656. It is not known whether G is irrational: this remains a stubbornly unsolved problem. The best hope for a solution might appear to be the method of Beukers [1] to prove the irrationality of ζ (2) directly from the series, but it is not clear how to adapt this method to G.

Type
Articles
Information
The Mathematical Gazette , Volume 101 , Issue 550 , March 2017 , pp. 38 - 49
Copyright
Copyright © Mathematical Association 2017 

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References

1. Beukers, F., A note on the irrationality of ζ (2) and ζ (3), Bull. London Math. Soc. 11 (1979) pp. 268–272.Google Scholar
2. Bradley, David M., Representations of Catalan's constant (2001), available at: http://researchgate.net/publication/2325473 Google Scholar
3. Adamchik, Victor, 33 representations for Catalan's constant, Carnegie Mellon University (2000) available at: http://www.cs.cmu.edu/~adamchik/articles/catalan Google Scholar
4. Lord, Nick, Intriguing integrals: an Euler-inspired odyssey, Math. Gaz. 91 (November 2007) pp. 415–427.Google Scholar
5. Boros, George and Moll, Victor H., Irresistible Integrals, Cambridge University Press (2004).Google Scholar
6. Lord, Nick, Variations on a theme – Euler and the logsine integral, Math. Gaz. 96 (November 2012) pp. 451–458.CrossRefGoogle Scholar
7. Scott, J. A., In praise of Catalan's constant, Math. Gaz. 86 (March 2002) pp. 102–103.Google Scholar
8. Math Overflow, Conjectured integral for Catalan's constant, accessed October 2016 at: http://mathoverflow.net/questions/181635 Google Scholar
9. Jameson, G. J. O., Elliptic integrals, the arithmetic-geometric mean and the Brent-Salamin algorithm for π, accessed October 2016 at http://www.maths.lancs.ac.uk/~jameson Google Scholar
10. Lord, Nick, Evaluating integrals using polar areas, Math. Gaz. 96 (July 2012), pp. 289–296.Google Scholar