sech

Sech, denoted sech(x), is the hyperbolic secant function. It is defined as sech(x) = 1 / cosh(x) = 2 / (e^x + e^{-x}). For real x, its domain is all real numbers.

Sech is an even function, since cosh is even, so sech(-x) = sech(x). Its range satisfies 0 < sech(x) ≤ 1, with sech(0) = 1 and sech(x) → 0 as |x| → ∞. As |x| grows, sech(x) decays roughly like 2 e^{-|x|}.

The derivative of sech is d/dx sech(x) = -sech(x) tanh(x), where tanh(x) = sinh(x)/cosh(x). An antiderivative is ∫ sech(x) dx = arcsin(tanh(x)) + C, and equivalently 2 arctan(tanh(x/2)) + C.

Inverse and related functions: on the interval where sech is monotone, its inverse is arcsech(y). The inverse is defined for 0 < y ≤ 1, with arcsech(y) ∈ [0, ∞). Related hyperbolic functions include cosh x = (e^x + e^{-x})/2 and tanh x = sinh x / cosh x, with the identity cosh^2 x − sinh^2 x = 1.

Applications and appearance: sech arises in various areas of mathematics and physics. It describes localized, bell-shaped profiles in soliton theory and optical pulse modeling, and it appears in certain probability and statistical contexts as a simple decaying function.