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A Catalan constant inspired integral odyssey

Published online by Cambridge University Press:  08 October 2020

Seán M. Stewart
Seán M. Stewart*
Affiliation:
9 Tanang Street, Bomaderry, NSW, 2541, Australia e-mail: sean.stewart@physics.org
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Extract

There is a rich and seemingly endless source of definite integrals that can be equated to or expressed in terms of Catalan's constant. Denoted by G and defined by

$${\rm{G}} = \sum\limits_{n = 0}^\infty {{{{{\left( { - 1} \right)}^n}} \over {{{\left( {2n + 1} \right)}^2}}} = 1 - {1 \over {{3^2}}} + {1 \over {{5^2}}} \ldots = 0.915\,965\,594 \ldots \,\,,} $$
Scott in [1] quipped that this constant seemed almost as useful as the more widely known Euler–Mascheroni constant γ, particularly in the evaluation of definite integrals. And like γ, Catalan's constant continues to remain one of the most inscrutable constants in mathematics where the question concerning its irrationality is not settled.

Type
Articles
Information
The Mathematical Gazette , Volume 104 , Issue 561 , November 2020 , pp. 449 - 459
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Copyright
© Mathematical Association 2020

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References

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