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A274440
Decimal expansion of Q(2), value of one of five integrals related to Quantum Field Theory (see the paper by David Broadhurst).
4
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
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
KEYWORD
nonn,cons
AUTHOR
STATUS
approved