A274440 Decimal expansion of Q(2), value of one of five integrals related to Quantum Field Theory (see the paper by David Broadhurst).

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

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_{x>0} arccosh((x+2)/2)^2 log((x+1)/x)/(x+n) dx.

Computation is done using the analytical form given by David Broadhurst:

Q(2) = -Cl2(Pi/3)^2 + 53/16 zeta(4) + 5/2 U, where Cl_2 is the Clausen integral and U is A255685.

Example

2.260399248120463689960929066240895031930761500163321388894889042329...

Mathematica

Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]);
U = A255685 = Pi^4/180 + (Pi^2/12)*Log[2]^2 - (1/12)*Log[2]^4 - 2*PolyLog[4, 1/2];
Q[2] = -Cl2[Pi/3]^2 + 53/16 Zeta[4] + 5/2 U;
RealDigits[N[Q[2], 104] // Chop][[1]]

Prog

(PARI)
Q(n) = intnum(x=0, oo, acosh((x+2)/2)^2 * log((x+1)/x)/(x+n));
Q(2) \\ Gheorghe Coserea, Sep 30 2018
(PARI)
clausen(n, x) = my(z = polylog(n, exp(I*x))); if (n%2, real(z), imag(z));
u31=Pi^4/180 + (Pi^2/12)*log(2)^2  - (1/12)*log(2)^4 - 2*polylog(4, 1/2);
-clausen(2, Pi/3)^2 + 53/16*zeta(4) + 5/2*u31 \\ Gheorghe Coserea, Sep 30 2018

Crossrefs

Cf. A274438 (Q(0)), A274439 (Q(1)), A274441 (Q(3)), A274442 (Q(4)).

Sequence in context: A091085 A011144 A127649 * A199220 A047916 A101207

Adjacent sequences: A274437 A274438 A274439 * A274441 A274442 A274443

Keyword

nonn,cons

Author

Jean-François Alcover, Jun 23 2016

Status

approved