A274441 -id:A274441 - cp's OEIS frontend

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

Jean-François Alcover, Jun 23 2016

			2.636185725224872226546402047919868685533952437408546504962614340...

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

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

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]]

Q(n) = intnum(x=0, oo, acosh((x+2)/2)^2 * log((x+1)/x)/(x+n));
Q(1) \\ Gheorghe Coserea, Sep 30 2018

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

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.

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

Original entry on oeis.org

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

Jean-François Alcover, Jun 23 2016

			2.260399248120463689960929066240895031930761500163321388894889042329...

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

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

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]]

Q(n) = intnum(x=0, oo, acosh((x+2)/2)^2 * log((x+1)/x)/(x+n));
Q(2) \\ Gheorghe Coserea, Sep 30 2018

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

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.

A274442 Decimal expansion of Q(4), value of one of five integrals related to Quantum Field Theory (see the paper by David Broadhurst).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 23 2016

			1.87447166949008260118095099948968029705739765892037953480769845119...

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

Cf. A274438 (Q(0)), A274439 (Q(1)), A274440 (Q(2)), A274441 (Q(3)).

digits = 101;
U = A255685 = Pi^4/180 + (Pi^2/12)*Log[2]^2 - (1/12)*Log[2]^4 - 2*PolyLog[4, 1/2];
v[k_] := ((-1)^k*((24*(k - 1)*(3*k - 4))/(3*k - 2)^3 + (8*(3*k*(3*k - 5) + 4))/(27*(k - 1)^3) + PolyGamma[2, (3*k)/2 - 1] - PolyGamma[2, (3*(k - 1))/2]))/(48*(k - 1)*(3*k - 4)*(3*k - 2));
V = A274400 = 3 Zeta[3]/8 - 1/2 + NSum[v[k], {k, 2, Infinity}, WorkingPrecision -> digits + 10, Method -> "AlternatingSigns"];
Q[4] = 125/54 Zeta[4] + 8 U - 8 V;
RealDigits[Q[4], 10, digits][[1]]

Q(n) = intnum(x=0, oo, acosh((x+2)/2)^2 * log((x+1)/x)/(x+n));
Q(4) \\ Gheorghe Coserea, Sep 30 2018

polygamma(n, x) = if (n == 0, psi(x), (-1)^(n+1)*n!*zetahurwitz(n+1, x));
u31=Pi^4/180 + (Pi^2/12)*log(2)^2  - (1/12)*log(2)^4 - 2*polylog(4, 1/2);
v31=3*zeta(3)/8 - 1/2 + sumalt(k=2, (-1)^k*((24*(k-1)*(3*k-4))/(3*k-2)^3 + (8*(3*k*(3*k-5)+4))/(27*(k-1)^3) + polygamma(2, (3*k)/2-1) - polygamma(2, (3*(k-1))/2))/(48*(k-1)*(3*k-4)*(3*k-2)));
125/54*zeta(4) + 8*u31 - 8*v31 \\ Gheorghe Coserea, Sep 30 2018

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(4) = 125/54 zeta(4) + 8 U - 8 V, where U is A255685 and V A274400.

A274438 Decimal expansion of Q(0), value of one of five integrals related to Quantum Field Theory (see the paper by David Broadhurst).

Original entry on oeis.org

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

Jean-François Alcover, Jun 23 2016

			4.1204258576856330093331932058655183968902289805100953379974262667755...

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

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

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]]

Q(n) = intnum(x=0, oo, acosh((x+2)/2)^2 * log((x+1)/x)/(x+n));
Q(0) \\ Gheorghe Coserea, Oct 01 2018

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

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.

cp's OEIS Frontend

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

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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).

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A274442 Decimal expansion of Q(4), value of one of five integrals related to Quantum Field Theory (see the paper by David Broadhurst).

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A274438 Decimal expansion of Q(0), value of one of five integrals related to Quantum Field Theory (see the paper by David Broadhurst).

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Mathematica

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