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

1,1

Links

Table of n, a(n) for n=1..104.

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

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.

Example

4.1204258576856330093331932058655183968902289805100953379974262667755...

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]]

Prog

(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
(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

Crossrefs

Cf. A274439 (Q(1)), A274440 (Q(2)), A274441 (Q(3)), A274442 (Q(4)).

Sequence in context: A343635 A287647 A331749 * A365387 A087565 A079163

Adjacent sequences: A274435 A274436 A274437 * A274439 A274440 A274441

Keyword

nonn,cons

Author

Jean-François Alcover, Jun 23 2016

Status

approved