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A274439 Decimal expansion of Q(1), value of one of five integrals related to Quantum Field Theory (see the paper by David Broadhurst). 5
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 (list; constant; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Jun 23 2016

STATUS

approved

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Last modified June 13 22:55 EDT 2021. Contains 345016 sequences. (Running on oeis4.)