A Catalan constant inspired integral odyssey

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}}}...

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Bibliographic Details
Published in:Mathematical gazette 2020-11, Vol.104 (561), p.449-459
Main Author: Stewart, Seán M.
Format: Article
Language:English
Subjects:
Integrals
Irrationality
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Summary: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.
ISSN:0025-5572
2056-6328
DOI:10.1017/mag.2020.99