arctangent integral

The inverse tangent integral Ti_2(x) is defined in terms of the dilogarithm Li_2(x) by Li_2(i x) = 1/4 Li_2(-x^2) + i Ti_2(x) (Lewin 1958, p. 33). It has the series Ti_2(x) = sum_(k = 1)^∞ (-1)^(k - 1) x^(2k - 1)/(2k - 1)^2 and gives in closed form the sum sum_(n = 1)^∞ (sin[(4n - 2) x])/(2n - 1)^2 = Ti_2(tan x) - x ln(tan x) that was considered by Ramanujan.

dilogarithm | L-algebraic number | Legendre's chi-function | Lerch transcendent | Rogers L-function