86.15 In Praise of the Catalan Constant

2002 ◽  
Vol 86 (505) ◽  
pp. 102 ◽  
Author(s):  
J. A. Scott
Keyword(s):  
Catalan Constant

scholarly journals A Double Inequality for the Trigamma Function and Its Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Zhen-Hang Yang ◽  
Yu-Ming Chu ◽  
Xiao-Jing Tao
Keyword(s):  
Approximation Algorithms ◽  
Double Inequality ◽  
Catalan Constant

We prove thatp=1andq=2are the best possible parameters in the interval(0,∞)such that the double inequalityep/x+1-e-p/x/2p<ψ′x+1<eq/x+1-e-q/x/2qholds forx>0. As applications, some new approximation algorithms for the circumference ratioπand Catalan constantG=∑n=0∞-1n/(2n+1)2are given. Here,ψ′is the trigamma function.

scholarly journals The Riemann zeta function and classes of infinite series

2017 ◽  
Vol 11 (2) ◽  
pp. 386-398 ◽  
Author(s):  
Horst Alzer ◽  
Junesang Choi
Keyword(s):  
Zeta Function ◽  
Infinite Series ◽  
Riemann Zeta Function ◽  
Catalan Constant ◽  
Series Representations ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Mathematical Constants

We present one-parameter series representations for the following series involving the Riemann zeta function ??n=3 n odd ?(n)/n sn and ??n=2 n even ?(n) n sn and we apply our results to obtain new representations for some mathematical constants such as the Euler (or Euler-Mascheroni) constant, the Catalan constant, log 2, ?(3) and ?.

scholarly journals THE BARNES G-FUNCTION AND THE CATALAN CONSTANT

2013 ◽  
Vol 67 (1) ◽  
pp. 105-116
Author(s):  
Xiaohan WANG
Keyword(s):  
Catalan Constant

A Catalan constant inspired integral odyssey

2020 ◽  
Vol 104 (561) ◽  
pp. 449-459
Author(s):  
Seán M. Stewart
Keyword(s):  
Definite Integrals ◽  
Catalan Constant ◽  
Catalan’S Constant ◽  
Image Position

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.

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