y = \coth x

Graph of the Hyperbolic Cotangent $y = \coth x$

The hyperbolic cotangent function $y = \coth x$ is defined as the ratio of the hyperbolic cosine to the hyperbolic sine, that is as the reciprocal of the hyperbolic tangent¹.

\coth x = \frac{\cosh x}{\sinh x} = \frac{1}{\tanh x} = \frac{e^x + e^{-x}}{e^x - e^{-x}}

It is the hyperbolic counterpart of the ordinary cotangent $\cot x$ .

Domain and range

The domain is $x \neq 0$
The range is $y < -1$ or $y > 1$
It decreases monotonically on each branch
It is an odd function

The denominator $\sinh x$ vanishes only at $x = 0$ . The value is always greater than $1$ or less than $-1$ , and nothing in $-1 \leq y \leq 1$ is ever taken.

Symmetry

Since $\coth(-x) = -\coth x$ , the function is odd and its graph is symmetric about the origin.

Monotonicity

The derivative is as follows.

\frac{d}{dx}\coth x = -\operatorname{csch}^2 x = 1 - \coth^2 x

It is negative throughout the domain, so the function is strictly decreasing on the branch with $x > 0$ and on the branch with $x < 0$ alike.

Asymptotes

Approach	$\coth x$
$x \to 0^{+}$	$\to +\infty$
$x \to 0^{-}$	$\to -\infty$
$x \to +\infty$	$\to 1$
$x \to -\infty$	$\to -1$

The line $x = 0$ is a vertical asymptote and the lines $y = 1$ and $y = -1$ are horizontal ones. The graph consists of two branches, one above and one below, separated by the origin.

Notable values

$x$	$\coth x$
$0.5$	$\approx 2.1640$
$1$	$\approx 1.3130$
$2$	$\approx 1.0373$
$3$	$\approx 1.0050$

Near the origin $\sinh x \approx x$ , so $\coth x \approx \dfrac{1}{x}$ and the divergence resembles that of an inverse proportion.

Comparison with the hyperbolic tangent

Item	$\tanh x$	$\coth x$
Domain	all real numbers	$x \neq 0$
Range	$(-1, 1)$	$\|y\| > 1$
Near the origin	$\approx x$	$\approx \dfrac{1}{x}$
Far out	$\to \pm 1$	$\to \pm 1$
Behavior	increasing	decreasing on each branch

Diverging near the origin and settling at $\pm 1$ far out is exactly the reverse of $\tanh x$ . The identity $\coth^2 x - \operatorname{csch}^2 x = 1$ also holds.

Applications

In statistical mechanics it appears as part of the Langevin function, which describes the magnetization of a paramagnet².

L(x) = \coth x - \frac{1}{x}

It also turns up in expressions related to Planck's law of black-body radiation and to Einstein's model of specific heat.

Hyperbolic functions, Wikipedia
Brillouin and Langevin functions, Wikipedia

Graph of the Hyperbolic Cotangent y=coth⁡xy = \coth xy=cothx