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

%I #9 May 20 2020 07:40:06

%S 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,

%T 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,

%U 0,6,1,0,2,4,3,5,5,6,0,9,0,4,4,7,2,2,6

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

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

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

%H Augustin-Louis Cauchy, <a href="https://archive.org/details/bub_gb_Hx0paIDkUbQC/page/n67/mode/2up">Mémoire sur les intégrales définies prises entre des limites imaginaires</a>, Paris, 1825, p. 65, equation 24.

%H Augustin-Louis Cauchy, <a href="https://archive.org/details/bub_gb_BmOo00FGCuAC/page/n115/mode/2up">Exercices de mathématiques</a>, Paris, 1826, p. 108, equation 48.

%H Edward Thomas Copson, An Introduction to the Theory of Functions of a Complex Variable, London, 1935, <a href="https://archive.org/details/in.ernet.dli.2015.233836/page/n163/mode/2up">Oxford, 1972 edition</a>, p. 153.

%H Paul J. Nahin, <a href="https://doi.org/10.1007/978-1-4939-1277-3">Inside Interesting Integrals</a>, Springer-Verlag New York, 2015, p. 340.

%H Cornel Ioan Vălean, <a href="https://doi.org/10.1007/978-3-030-02462-8">(Almost) Impossible Integrals, Sums, and Series</a>, Springer International Publishing, 2019, p. 206-207.

%F Equals Integral_{x=0..oo} (exp(cos(x)) * sin(sin(x))/x) * dx (Cauchy, 1825-26).

%F Equals Integral_{x=0..oo} (exp(cos(x)) * sin(x) * sin(sin(x))/x^2) * dx (Vălean, 2019).

%F Equals A019610 - A019669.

%e 2.69907078454188691350045374313353580541885956819500...

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

%o (PARI) Pi*(exp(1)-1)/2 \\ _Michel Marcus_, May 20 2020

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

%K nonn,cons

%O 1,1

%A _Amiram Eldar_, May 20 2020

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