Graph of the Hyperbolic Secant $y = \operatorname{sech} x$

The hyperbolic secant function $y = \operatorname{sech} x$ is defined as the reciprocal of the hyperbolic cosine:

It is the hyperbolic analogue of the secant $\sec x$.

Domain and range

Since $\cosh x \geq 1$ is never zero, $\operatorname{sech} x$ is defined for every real $x$, so its domain is $(-\infty, \infty)$. From $\cosh x \geq 1$ we get $0 < \operatorname{sech} x \leq 1$, so its range is the half-open interval $(0, 1]$; the values are positive and never reach $0$.

Symmetry

Because $\cosh x$ is even, its reciprocal $\operatorname{sech} x$ is also even, and its graph is symmetric about the $y$-axis.

Monotonicity and maximum

The derivative is $\dfrac{d}{dx}\operatorname{sech} x = -\operatorname{sech} x \tanh x$. It is positive for $x < 0$ (where $\tanh x < 0$) and negative for $x > 0$. Thus $\operatorname{sech} x$ increases on $(-\infty, 0)$ and decreases on $(0, \infty)$, reaching its maximum at $(0, 1)$.

Asymptotes and limits

As $x \to \pm\infty$, $\cosh x \to \infty$, so $\operatorname{sech} x \to 0$. The $x$-axis (the line $y = 0$) is therefore a horizontal asymptote, approached from above since the values stay positive.

Notable points

The curve passes through its peak $(0, 1)$, with $\operatorname{sech} 0 = 1$. It is a symmetric bell-shaped curve that rises in the middle and decays smoothly on both sides. It resembles the normal distribution's density, though its tails decay exponentially.

Relationships with other functions

By definition $\operatorname{sech} x = \dfrac{1}{\cosh x}$. Dividing the identity $\cosh^2 x - \sinh^2 x = 1$ by $\cosh^2 x$ gives $1 - \tanh^2 x = \operatorname{sech}^2 x$, which equals the derivative of $\tanh x$.

Applications

The smooth bell shape of $\operatorname{sech} x$ appears in the soliton solutions of nonlinear wave equations such as the KdV equation and the nonlinear Schrodinger equation. Solitary-wave profiles take the form $\operatorname{sech}$ or $\operatorname{sech}^2$, making the function important for modeling optical pulses in fibers and waves in shallow water.