A019610 -id:A019610 - cp's OEIS frontend

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

Jean-François Alcover, Aug 08 2014

			1.54912659257756216836957253384940992693702986341004836289999671...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin Constant, p. 136.

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

Cf. A006752, A019610 (D(2)), A074962, A241421 (D(1)).

(E^(Catalan/Pi)*Glaisher^3*Sqrt[Gamma[3/4]/Gamma[1/4]])/2^(1/12) // RealDigits[#, 10, 104]& // First

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

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.

A241421 Decimal expansion of D(1), where D(x) is the infinite product function defined in the formula section (or in the Finch reference).

Original entry on oeis.org

2, 2, 3, 5, 8, 8, 5, 5, 9, 5, 5, 0, 8, 9, 6, 9, 8, 6, 4, 2, 8, 3, 9, 6, 4, 7, 9, 9, 3, 1, 1, 8, 9, 0, 6, 4, 4, 8, 4, 5, 1, 5, 9, 1, 2, 2, 8, 5, 9, 5, 2, 4, 7, 4, 7, 7, 9, 3, 4, 4, 7, 9, 7, 8, 2, 6, 0, 6, 2, 7, 0, 8, 1, 4, 5, 7, 2, 5, 2, 2, 1, 7, 9, 3, 2, 8, 3, 2, 0, 2, 9, 5, 2, 8, 3, 2, 3, 4, 6, 2, 8, 9, 8, 2

Offset: 1

Jean-François Alcover, Aug 08 2014

			2.23588559550896986428396479931189064484515912285952474779344797826...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin Constant, p. 136.

G. C. Greubel, Table of n, a(n) for n = 1..5000
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, Glaisher-Kinkelin Constant

Cf. A006752, A019610 (D(2)), A074962, A241420 (D(1/2)).

RealDigits[Glaisher^6/(2^(1/6)*Sqrt[Pi]), 10, 104] // First

default(realprecision, 100); A=exp(1/12-zeta'(-1)); A^6/(2^(1/6)* sqrt(Pi)) \\ G. C. Greubel, Aug 24 2018

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) = A^6/(2^(1/6)*sqrt(Pi)).

A335027 Decimal expansion of Pi*(e-1)/2.

Original entry on oeis.org

2, 6, 9, 9, 0, 7, 0, 7, 8, 4, 5, 4, 1, 8, 8, 6, 9, 1, 3, 5, 0, 0, 4, 5, 3, 7, 4, 3, 1, 3, 3, 5, 3, 5, 8, 0, 5, 4, 1, 8, 8, 5, 9, 5, 6, 8, 1, 9, 5, 0, 0, 4, 5, 7, 0, 4, 5, 2, 3, 2, 8, 2, 6, 8, 9, 3, 5, 7, 0, 6, 1, 0, 2, 4, 3, 5, 5, 6, 0, 9, 0, 4, 4, 7, 2, 2, 6

Offset: 1

Amiram Eldar, May 20 2020

The value of an integral (see formula) first calculated by Cauchy in 1825 (with an error that was corrected in 1826).

This integral appears in the forward to Vălean's book, written by Paul J. Nahin.

			2.69907078454188691350045374313353580541885956819500...

Augustin-Louis Cauchy, Mémoire sur les intégrales définies prises entre des limites imaginaires, Paris, 1825, p. 65, equation 24.
Augustin-Louis Cauchy, Exercices de mathématiques, Paris, 1826, p. 108, equation 48.
Edward Thomas Copson, An Introduction to the Theory of Functions of a Complex Variable, London, 1935, Oxford, 1972 edition, p. 153.
Paul J. Nahin, Inside Interesting Integrals, Springer-Verlag New York, 2015, p. 340.
Cornel Ioan Vălean, (Almost) Impossible Integrals, Sums, and Series, Springer International Publishing, 2019, p. 206-207.

Cf. A000796 (Pi), A001113 (e), A019609 (Pi*e), A019610(Pi*e/2), A019669 (Pi/2), A335028.

Mathematica
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RealDigits[Pi*(E-1)/2, 10, 100][[1]]
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Pi*(exp(1)-1)/2 \\ Michel Marcus, May 20 2020

Equals Integral_{x=0..oo} (exp(cos(x)) * sin(sin(x))/x) * dx (Cauchy, 1825-26).
Equals Integral_{x=0..oo} (exp(cos(x)) * sin(x) * sin(sin(x))/x^2) * dx (Vălean, 2019).
Equals A019610 - A019669.

A335028 Decimal expansion of Pi*(exp(1/e) - 1)/2.

Original entry on oeis.org

6, 9, 8, 4, 8, 2, 6, 4, 2, 7, 1, 7, 8, 8, 4, 2, 7, 2, 2, 6, 7, 2, 3, 0, 3, 5, 8, 4, 9, 7, 7, 1, 2, 4, 4, 4, 5, 6, 2, 8, 4, 8, 3, 6, 6, 9, 3, 2, 9, 2, 9, 7, 9, 3, 6, 9, 9, 3, 7, 2, 3, 6, 6, 2, 3, 3, 4, 5, 9, 0, 3, 0, 1, 2, 5, 4, 3, 6, 9, 0, 4, 3, 0, 0, 6, 9, 8

Offset: 0

Amiram Eldar, May 20 2020

The value of an integral (see formula) first calculated by Cauchy in 1825 (with an error that was corrected in 1826).

			0.69848264271788427226723035849771244456284836693292...

Harold P. Boas, Cauchy’s Residue Sore Thumb, The American Mathematical Monthly, Vol. 125, No. 1 (2018), pp. 16-28, preprint, arXiv:1701.04887v1 [math.HO], 2017.
Augustin-Louis Cauchy, Mémoire sur les intégrales définies prises entre des limites imaginaires, Paris, 1825, p. 65, equation 24.
Augustin-Louis Cauchy, Exercices de mathématiques, Paris, 1826, p. 108, equation 48.

Cf. A000796 (Pi), A001113 (e), A019609 (Pi*e), A019610(Pi*e/2), A073229 (e^(1/e)), A335027.

RealDigits[Pi*(Exp[1/E] - 1)/2, 10, 100][[1]]

Pi*(exp(1/exp(1)) - 1)/2 \\ Michel Marcus, May 20 2020

Equals Integral_{x=0..oo} (exp(cos(x)) * sin(sin(x)) * x /(x^2 + 1)) * dx.

A094936 Decimal expansion of 2/(Pi*e).

Original entry on oeis.org

2, 3, 4, 1, 9, 9, 3, 2, 6, 0, 9, 7, 2, 7, 6, 6, 4, 2, 7, 6, 0, 9, 6, 9, 0, 7, 3, 8, 6, 6, 6, 6, 7, 4, 8, 8, 8, 5, 5, 6, 5, 9, 6, 8, 5, 1, 0, 4, 2, 4, 5, 7, 9, 5, 5, 0, 7, 8, 8, 9, 0, 4, 3, 8, 8, 0, 6, 5, 1, 3, 8, 7, 0, 7, 4, 8, 7, 8, 2, 6, 1, 5, 3, 2, 0, 7, 5, 0, 4, 4, 0, 3, 6, 2, 3, 3, 4, 5, 7, 2, 6, 2, 6, 2

Offset: 0

Benoit Cloitre, Jun 18 2004

If N(n) denotes the number of real roots of the n-th Bernoulli polynomial then lim_{n->oo} N(n)/n = 2/(e*Pi).

			0.23419932609727664276...

A. P. Veselov and J.P. Ward, On the real roots of the Bernoulli polynomials and the Hurwitz zeta-function, 1999.
A. P. Veselov and J.P. Ward, On the real zeroes of the Hurwitz zeta-function and Bernoulli polynomials, J. Math. Anal. Appl. 305 (2) (2005) 712-721.

Cf. A000796 (Pi), A001113 (e), A019609, A019610.

RealDigits[2/(Pi*E), 10, 120][[1]] (* Amiram Eldar, Apr 28 2025 *)

2/exp(1)/Pi \\ Charles R Greathouse IV, Jan 14 2017

Equals 1/A019610. - Michel Marcus, Dec 07 2013

cp's OEIS Frontend

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).

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A241421 Decimal expansion of D(1), where D(x) is the infinite product function defined in the formula section (or in the Finch reference).

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A335027 Decimal expansion of Pi*(e-1)/2.

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A335028 Decimal expansion of Pi*(exp(1/e) - 1)/2.

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A094936 Decimal expansion of 2/(Pi*e).

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