Inverse Tangent Integral

The inverse tangent integral, written Ti2 of x, is a special function defined by integrating the arctangent of t divided by t from zero to x, using the principal branch of the arctangent. It can also be expressed as the alternating power series x minus x cubed over three squared plus x to the fifth over five squared and so on, which converges for every x with absolute value at most one. The function is closely tied to the dilogarithm, another special function studied in analysis: Ti2 of x equals the imaginary part of the dilogarithm evaluated at i times x. At x equal to one, the inverse tangent integral equals Catalan's constant, approximately 0.916, and the function is odd, meaning Ti2 of negative x equals negative Ti2 of x. The notation Ti2 was introduced by the mathematician Lewin, but the function itself had already been studied earlier by William Spence in 1809 and later by Srinivasa Ramanujan. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/