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A254135
Decimal expansion of Lamb's integral K_2.
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
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
KEYWORD
nonn,cons,easy
AUTHOR
STATUS
approved