sech2x

Sech2x is commonly used to denote the square of the hyperbolic secant function, written as sech^2 x. Since the hyperbolic secant is defined by sech x = 1 / cosh x and cosh x = (e^x + e^{-x})/2, it follows that sech^2 x = 1 / cosh^2 x = 4 / (e^x + e^{-x})^2.

Sech^2 x is defined for all real x and is an even function, symmetric about the origin. Its range is (0, 1], attaining its maximum value of 1 at x = 0 and decreasing toward 0 as |x| grows. It is related to the hyperbolic tangent by the identity sech^2 x = 1 − tanh^2 x, since tanh x = sinh x / cosh x.

Calculus and differentiation play a central role with this function. The derivative is d/dx [sech^2 x] = −2 sech^2 x tanh x. The derivative of the hyperbolic tangent is sech^2 x, i.e., d/dx tanh x = sech^2 x, which explains why the antiderivative of sech^2 x is tanh x: ∫ sech^2 x dx = tanh x + C.

A compact exponential form is sech^2 x = 4 / (e^x + e^{-x})^2, which can be useful for certain computations.

Applications of sech^2 x appear in differential equations and mathematical physics; notably, the Korteweg–de Vries equation has soliton solutions of the form u(x,t) = sech^2((x − ct)/2). For contrasts or alternative uses, sech(2x) denotes the hyperbolic secant of 2x, i.e., 1 / cosh(2x), which is a different function.