A274438 Decimal expansion of Q(0), value of one of five integrals related to Quantum Field Theory (see the paper by David Broadhurst).
4, 1, 2, 0, 4, 2, 5, 8, 5, 7, 6, 8, 5, 6, 3, 3, 0, 0, 9, 3, 3, 3, 1, 9, 3, 2, 0, 5, 8, 6, 5, 5, 1, 8, 3, 9, 6, 8, 9, 0, 2, 2, 8, 9, 8, 0, 5, 1, 0, 0, 9, 5, 3, 3, 7, 9, 9, 7, 4, 2, 6, 2, 6, 6, 7, 7, 5, 5, 4, 4, 1, 5, 8, 1, 0, 1, 0, 7, 0, 2, 6, 0, 8, 9, 2, 0, 1, 6, 3, 9, 2, 6, 8, 5, 9, 1, 6, 4, 5, 3, 9, 8, 2, 9
Offset: 1
Examples
4.1204258576856330093331932058655183968902289805100953379974262667755...
Links
- David J. Broadhurst, Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity, arXiv:hep-th/9803091, 1998, p. 12.
- Eric Weisstein's MathWorld, Clausen's Integral
Programs
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Mathematica
Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]); Q[0] = 4 Cl2[Pi/3]^2 ; RealDigits[N[Q[0], 104] // Chop][[1]]
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PARI
Q(n) = intnum(x=0, oo, acosh((x+2)/2)^2 * log((x+1)/x)/(x+n)); Q(0) \\ Gheorghe Coserea, Oct 01 2018
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PARI
clausen(n, x) = my(z = polylog(n, exp(I*x))); if (n%2, real(z), imag(z)); 4*clausen(2, Pi/3)^2 \\ Gheorghe Coserea, Oct 01 2018
Formula
Q(n) = Integral_{0..inf} arccosh((x+2)/2)^2 log((x+1)/x)/(x+n) dx.
Computation is done using the analytical form given by David Broadhurst: Q(0) = 4 Cl_2(Pi/3)^2, where Cl_2 is the Clausen integral.
15 Q(0) + 144 Q(1) - 448 Q(2) + 126 Q(3) + 168 Q(4) = 0.