A254135 Decimal expansion of Lamb's integral K_2.

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

0,1

Links

Table of n, a(n) for n=0..102.

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.

Formula

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)).

Example

0.69266081515264750650943118588427245846713483280766884258...

Maple

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

Mathematica

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

Crossrefs

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

Sequence in context: A309825 A289503 A349522 * A198676 A198616 A215668

Adjacent sequences: A254132 A254133 A254134 * A254136 A254137 A254138

Keyword

nonn,cons,easy,changed

Author

Jean-François Alcover, Jan 26 2015

Status

approved