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

0,1

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_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)).

Example

0.490772728958345159162717253203382640381923347758584656...

Maple

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

Mathematica

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

Crossrefs

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

Sequence in context: A243372 A020802 A085675 * A303984 A336308 A070439

Adjacent sequences: A254130 A254131 A254132 * A254134 A254135 A254136

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Jan 26 2015

Status

approved