A274439 Decimal expansion of Q(1), value of one of five integrals related to Quantum Field Theory (see the paper by David Broadhurst).

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

Offset

1,1

Links

Table of n, a(n) for n=1..103.

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(1) = (4/3)*Cl2(Pi/3)^2 + (7/6)*zeta(4), where Cl_2 is the Clausen integral.

Example

2.636185725224872226546402047919868685533952437408546504962614340...

Mathematica

Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]);
Q[1] = 4/3 Cl2[Pi/3]^2 + 7/6 Zeta[4];
RealDigits[N[Q[1], 103] // Chop][[1]]

Prog

(PARI)
Q(n) = intnum(x=0, oo, acosh((x+2)/2)^2 * log((x+1)/x)/(x+n));
Q(1) \\ Gheorghe Coserea, Sep 30 2018
(PARI)
clausen(n, x) = my(z = polylog(n, exp(I*x))); if (n%2, real(z), imag(z));
4/3*clausen(2, Pi/3)^2 + 7/6*zeta(4) \\ Gheorghe Coserea, Sep 30 2018

Crossrefs

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

Sequence in context: A139384 A083481 A177209 * A280342 A275476 A185380

Adjacent sequences: A274436 A274437 A274438 * A274440 A274441 A274442

Keyword

nonn,cons

Author

Jean-François Alcover, Jun 23 2016

Status

approved