sech2

Sech squared, written as sech^2 x or (sech x)^2, denotes the square of the hyperbolic secant function. The hyperbolic secant is defined by sech x = 1 / cosh x, so sech^2 x = 1 / cosh^2 x. The function is real-valued for all real x and is even, positive everywhere, with a maximum value of 1 at x = 0. As |x| increases, sech^2 x decays exponentially, with asymptotic form sech^2 x ~ 4 e^{-2|x|} for large |x|, due to cosh x ~ (1/2) e^{|x|}.

Key properties include its derivative and integral. The derivative is d/dx sech^2 x = -2 sech^2 x tanh x, while the antiderivative is ∫ sech^2 x dx = tanh x + C. A fundamental identity is sech^2 x = 1 − tanh^2 x, derived from 1 − tanh^2 x = sech^2 x and the relation tanh x = sinh x / cosh x. The derivative of tanh x is sech^2 x, establishing a close link between the two functions.

In the complex plane, sech x is analytic except at the zeros of cosh x, which occur at x = i(π/2 + πk); sech^2 x has double poles at these points. The function arises in various contexts: in soliton theory, the squared hyperbolic secant profile describes one-soliton solutions to certain nonlinear wave equations; in quantum mechanics, the Pöschl–Teller potential involves a sech^2 form and is exactly solvable. In optics and signal processing, sech^2 envelopes model localized pulses and wave packets.

See also hyperbolic functions, tanh, and related solvable potentials.