A244922 -id:A244922 - cp's OEIS frontend

A254133 Decimal expansion of Lamb's integral K_0.

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

Offset: 0

Jean-François Alcover, Jan 26 2015

			0.490772728958345159162717253203382640381923347758584656...

D. H. Bailey, J. M. Borwein, and R. E. Crandall, Advances in the theory of box integrals (2010) p. 18.
Eric Weisstein's World of Mathematics, Inverse Tangent Integral.

Cf. A244920, A244921, A244922, A254134, A254135.

evalf(int(arctanh(1/sqrt(3 + x^2))/(1 + x^2), x=0..1), 120); # Vaclav Kotesovec, Jan 26 2015

Ti2[x_] := (I/2)*(PolyLog[2, -I*x] - PolyLog[2, I*x]); K0 = (3/2)*Ti2[3 - 2 Sqrt[2]] + Pi/4*Log[1 + Sqrt[2]] - Catalan/2 // Re; RealDigits[K0, 10, 103] // First

K_0 = integral_[0..1] arctanh(1/sqrt(3 + x^2))/(1 + x^2) dx.

K_0 = 3/2*Ti_2(3 - 2*sqrt(2)) + Pi/4*log(1 + sqrt(2)) - Catalan/2, where Ti_2 is Lewin's arctan integral, Ti_2(x) = (i/2)*(Li_2(-i*x) - Li_2(i*x)).

A254134 Decimal expansion of Lamb's integral K_1.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jan 26 2015

			1.6619078747381233774065816861630594973488686732512589...

D. H. Bailey, J. M. Borwein, R. E. Crandall, Advances in the theory of box integrals (2010) p. 18.
Eric Weisstein's MathWorld, Clausen's Integral

Cf. A244920, A244921, A244922, A254133, A254135.

evalf(int(arcsec(x)/sqrt(x^2 - 4*x + 3), x=3..4), 120); # Vaclav Kotesovec, Jan 26 2015

Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]); th = (ArcTan[(16 - 3*Sqrt[15])/11] + Pi)/3; K1 = Cl2[th] - Cl2[th + Pi/3] - Cl2[th - Pi/2] + Cl2[th - Pi/6] - Cl2[3*th + Pi/3] + Cl2[3*th + 2*(Pi/3)] - Cl2[3*th - 5*(Pi/6)] + Cl2[3*th + 5*(Pi/6)] + (6*th - 5*(Pi/2))*Log[2 - Sqrt[3]] // Re; RealDigits[K1, 10, 103] // First

K_1 = integral_[3..4] arcsec(x)/sqrt(x^2 - 4*x + 3) dx.

K_1 = Cl_2(th) - Cl_2(th + Pi/3) - Cl_2(th - Pi/2) + Cl_2(th - Pi/6) - Cl_2(3*th + Pi/3) + Cl_2(3*th + 2*(Pi/3)) - Cl_2(3*th - 5*(Pi/6)) + Cl_2(3*th + 5*(Pi/6)) + (6*th - 5*(Pi/2))*log(2 - sqrt(3)), where Cl_2 is the Clausen function and th = (arctan((16 - 3*sqrt(15))/11) + Pi)/3.

A254135 Decimal expansion of Lamb's integral K_2.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jan 26 2015

			0.69266081515264750650943118588427245846713483280766884258...

D. H. Bailey, J. M. Borwein, and R. E. Crandall, Advances in the theory of box integrals (2010) p. 18.
Eric Weisstein's World of Mathematics, Inverse Tangent Integral.

Cf. A244920, A244921, A244922, A254133, A254134.

evalf(int(sqrt(1 + sec(x)^2)*arctan(1/sqrt(1 + sec(x)^2)), x=0..Pi/4), 120); # Vaclav Kotesovec, Jan 26 2015

Ti2[x_] := (I/2)* (PolyLog[2, -I *x] - PolyLog[2, I *x]); K2 = (1/2)*Ti2[-2 + Sqrt[3]] + (Pi/8)*Log[2 + Sqrt[3]] + Pi^2/32 // Re; RealDigits[K2, 10, 103] // First

K_2 = integral_[0..Pi/4] sqrt(1 + sec(x)^2)*arctan(1/sqrt(1 + sec(x)^2)) dx.

K_2 = (1/2)*Ti_2(-2 + sqrt(3)) + (Pi/8)*log(2 + sqrt(3)) + Pi^2/32, where Ti_2 is Lewin's arctan integral, Ti_2(x) = (i/2)*(Li_2(-i*x) - Li_2(i*x)).

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A254133 Decimal expansion of Lamb's integral K_0.

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A254134 Decimal expansion of Lamb's integral K_1.

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A254135 Decimal expansion of Lamb's integral K_2.

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