A254134 Decimal expansion of Lamb's integral K_1.

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

Author

Jean-François Alcover, Jan 26 2015

Keywords

• nonn
• cons
• easy

Examples

			1.6619078747381233774065816861630594973488686732512589...
		

Links

• 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

Crossrefs

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

Programs

Maple

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

Mathematica

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

Formula

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.