A241420 Decimal expansion of D(1/2), where D(x) is the infinite product function defined in the formula section (or in the Finch reference).
1, 5, 4, 9, 1, 2, 6, 5, 9, 2, 5, 7, 7, 5, 6, 2, 1, 6, 8, 3, 6, 9, 5, 7, 2, 5, 3, 3, 8, 4, 9, 4, 0, 9, 9, 2, 6, 9, 3, 7, 0, 2, 9, 8, 6, 3, 4, 1, 0, 0, 4, 8, 3, 6, 2, 8, 9, 9, 9, 9, 6, 7, 1, 0, 3, 9, 9, 8, 3, 8, 0, 0, 8, 3, 6, 5, 4, 3, 2, 9, 8, 7, 4, 0, 6, 5, 1, 1, 4, 0, 9, 2, 0, 7, 0, 0, 8, 0, 6, 1, 5, 4, 6, 4
Offset: 1
Examples
1.54912659257756216836957253384940992693702986341004836289999671...
References
- Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin Constant, p. 136.
Links
- G. C. Greubel, Table of n, a(n) for n = 1..10000
- L. Almodovar, V. H. Moll, H. Quand, Infinite products arising in paperfolding, JIS 19 (2016) # 16.5.1 eq. (15).
- Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 20.
- Eric Weisstein's MathWorld, Barnes G-Function
- Eric Weisstein's MathWorld, Catalan's Constant
- Eric Weisstein's MathWorld, Glaisher-Kinkelin Constant
Programs
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Mathematica
(E^(Catalan/Pi)*Glaisher^3*Sqrt[Gamma[3/4]/Gamma[1/4]])/2^(1/12) // RealDigits[#, 10, 104]& // First
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PARI
default(realprecision, 100); A=exp(1/12-zeta'(-1)); exp(Catalan/Pi)*A^3*sqrt(gamma(3/4)/gamma(1/4))/2^(1/12) \\ G. C. Greubel, Aug 24 2018
Formula
D(x) = lim_{n->infinity} ( Product_{k=1..2n+1} (1+x/k)^((-1)^(k+1)*k) ).
D(x) = (e^(x/2-1/4)*A^3*G((x+1)/2)^2*Gamma(x/2)^(x-2)*Gamma((x+1)/2)^(1-x)*(Gamma((x+1)/2)/Gamma(x/2))^x)/(2^(1/12)*G(x/2)^2), where A is the Glaisher-Kinkelin constant and G is the Barnes G-function.
D(1/2) = (e^(C/Pi)*A^3*sqrt(Gamma(3/4)/Gamma(1/4)))/2^(1/12), where C is Catalan's constant.