sechx

Sech x, written as sech x, is the hyperbolic secant function. It is defined by sech x = 1 / cosh x, with cosh x = (e^x + e^-x)/2. Equivalently, sech x = 2 / (e^x + e^-x). The function is even and positive for all real x, attaining its maximum value of 1 at x = 0 and tending to 0 as |x| grows.

Basic properties include its derivatives and relationships with other hyperbolic functions. The derivative is d/dx sech x = -sech x tanh x. The second derivative is sech x (2 tanh^2 x - 1). A key identity is sech^2 x = 1 - tanh^2 x. These relations link sech to tanh and sinh.

Integrals of sech x have standard forms. One common antiderivative is ∫ sech x dx = arctan(sinh x) + C. Equivalently, ∫ sech x dx = 2 arctan(tanh(x/2)) + C.

Applications and appearances. Sech x arises in solving certain differential equations and in Fourier analysis, where the Fourier transform of sech x is π sech(π ω / 2) under the common convention. In probability, the hyperbolic secant distribution uses a density proportional to sech(π x / 2), specifically f(x) = (1/2) sech(π x / 2), as a continuous, symmetric distribution with finite variance.

In summary, sech x is a simple, well-behaved function defined via cosh, notable for its even symmetry, simple derivative, exact antiderivative, and appearances in analysis and probability.