The functions E_1(x) | = | (x^2 e^x)/(e^x - 1)^2 E_2(x) | = | x/(e^x - 1) E_3(x) | = | ln(1 - e^(-x)) E_4(x) | = | x/(e^x - 1) - ln(1 - e^(-x)). E_1(x) has an inflection point at E_1^, , (x) = 1/8 csch^4(1/2 x)[(x^2 + 2) cosh x + 2(x^2 - 2x sinh x - 1)] = 0, which can be solved numerically to give x = ± 2.34694... (OEIS A118080).

Bose-Einstein distribution | Fermi-Dirac distribution | sigmoid function