

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
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,1


LINKS



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



AUTHOR



STATUS

approved



