All functions are normalized in such a way that they satisfy

• −1 < ƒ(x) < 1
• ƒ(−x) = −ƒ(x) and hence ƒ(0) = 0
• 0 < ƒ'(x) ≤ 1 and ƒ'(x) = 1 iff x = 0
• ƒ(∞) = 1 and ƒ(−∞) = −1
• all functions are analytic except x/(1+|x|) which is not analytic at 0.

For notations see also

• inverse trigonometric functions (arctan)
• error function (erf)
• gudermannian function (gd)
• hyperbolic function (tanh)