A247685 Decimal expansion of the integral over the square (0,1)x(0,1) of 1/((x+y)*sqrt((1-x)*(1-y))) dx dy.
3, 6, 6, 3, 8, 6, 2, 3, 7, 6, 7, 0, 8, 8, 7, 6, 0, 6, 0, 2, 1, 8, 4, 1, 4, 0, 5, 9, 7, 2, 9, 5, 3, 6, 4, 4, 3, 0, 9, 6, 5, 9, 7, 4, 9, 7, 1, 2, 6, 6, 8, 8, 5, 3, 7, 0, 6, 5, 9, 9, 2, 4, 7, 8, 4, 8, 7, 0, 5, 2, 0, 7, 9, 1, 0, 5, 0, 1, 9, 0, 7, 7, 9, 1, 7, 4, 2, 6, 0, 5, 1, 7, 0, 4, 4, 6, 0, 4, 2, 4, 9, 9, 4
Offset: 1
Examples
3.663862376708876060218414059729536443096597497126688537...
Links
- G. C. Greubel, Table of n, a(n) for n = 1..10000
- Ian Agol, The minimal volume orientable hyperbolic 2-cusped 3-manifolds, Proc. Amer. Math. Soc., Vol. 138, No. 10 (2010), pp. 3723-3732. MR2661571
- David H. Bailey and Jonathan M. Borwein, Highly Parallel, High-Precision Numerical Integration, Lawrence Berkeley National Laboratory, 2005, p. 9.
- Eric Weisstein's World of Mathematics, Lerch Transcendent.
- Wikipedia, Lerch zeta function.
Programs
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Magma
SetDefaultRealField(RealField(100)); R:=RealField(); 4*Catalan(R); // G. C. Greubel, Aug 25 2018
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Maple
evalf(4*Catalan, 130); # Alois P. Heinz, Aug 14 2023
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Mathematica
RealDigits[4*Catalan, 10, 103] // First
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PARI
default(realprecision, 100); 4*Catalan \\ G. C. Greubel, Aug 25 2018
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PARI
lerchphi(-1, 2, 1/2) \\ Charles R Greathouse IV, Jan 30 2025
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PARI
sumalt(k=0, (-1)^k/(k+1/2)^2) \\ Charles R Greathouse IV, Jan 30 2025
Formula
Equals 4*Catalan.
Equals Integral_{x=0..Pi/2} log((1+cos(x))/(1-cos(x))) dx = Integral_{x=0..Pi/2} log((1+sin(x))/(1-sin(x))) dx. - Amiram Eldar, Apr 07 2022
From Amiram Eldar, Aug 14 2023: (Start)
Equals Phi(-1, 2, 1/2) = Sum_{k>=0} (-1)^k/(k+1/2)^2, where Phi is the Lerch transcendent.
Equals Integral_{x=-Pi/2..Pi/2} x/sin(x) dx. (End)
Comments