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 cosine¹.

It is the hyperbolic counterpart of the ordinary secant $\sec x$.

Domain and range

• The domain is all real numbers
• The range is the half-open interval $(0, 1]$
• The maximum $1$ is attained at $x = 0$
• It is an even function

The denominator $\cosh x$ is always at least $1$ and never $0$, so the function is defined for every real number. Since $\cosh x \geq 1$, the value stays at $1$ or below and never reaches $0$. Where $\sec x$ had asymptotes and satisfied $|y| \geq 1$, this function is the exact opposite: bounded and smooth.

Symmetry

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

Monotonicity and maximum

The derivative is $\dfrac{d}{dx}\operatorname{sech} x = -\operatorname{sech} x \tanh x$. As $\operatorname{sech} x$ is positive, the sign comes from $-\tanh x$: the function increases for $x < 0$ and decreases for $x > 0$. The maximum point is $(0, 1)$.

Asymptote and decay

As $x \to \pm\infty$ we have $\cosh x \to \infty$ and hence $\operatorname{sech} x \to 0$, so the $x$-axis is a horizontal asymptote. The values are always positive, so the curve approaches $0$ from above.

Far out $\cosh x \approx \dfrac{e^{|x|}}{2}$, so the decay is exponential.

The graph is a bell shape, symmetric, raised in the middle and falling away smoothly on both sides. It resembles the density of a normal distribution, but its tails decay exponentially rather than as steeply as a Gaussian.

An identity

Dividing both sides of $\cosh^2 x - \sinh^2 x = 1$ by $\cosh^2 x$ gives the following.

The right-hand side is exactly the derivative of $\tanh x$. It corresponds to $1 + \tan^2 x = \sec^2 x$ for trigonometric functions, differing only in a sign.

Applications

The smooth bell shape appears in the soliton solutions of equations describing nonlinear waves². The profile of a solitary wave is written with $\operatorname{sech}$ or $\operatorname{sech}^2$.

• Solitons of the KdV equation and of the nonlinear Schrödinger equation
• Light pulses that travel along an optical fiber without changing shape
• Solitary waves on shallow water

1. Hyperbolic functions, Wikipedia
2. Soliton, Wikipedia