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, the reciprocal of the hyperbolic tangent:

It is the hyperbolic analogue of the cotangent $\cot x$.

Domain and range

The denominator $\sinh x$ vanishes only at $x = 0$, so $\coth x$ is defined for all real $x \neq 0$. Its values are always greater than $1$ or less than $-1$, so its range is $(-\infty, -1) \cup (1, \infty)$, i.e. $|y| > 1$; it never takes values in $[-1, 1]$.

Symmetry

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

Monotonicity

The derivative is $\dfrac{d}{dx}\coth x = -\operatorname{csch}^2 x = 1 - \coth^2 x$, which is negative throughout the domain. Hence $\coth x$ is strictly decreasing on each of its two branches, for $x > 0$ and for $x < 0$.

Asymptotes and limits

The line $x = 0$ is a vertical asymptote: as $x \to 0^+$, $\coth x \to +\infty$, and as $x \to 0^-$, $\coth x \to -\infty$. As $x \to +\infty$, $\coth x \to 1$, and as $x \to -\infty$, $\coth x \to -1$, so $y = 1$ and $y = -1$ are horizontal asymptotes. The graph consists of two separate branches on either side of the origin.

Notable points

Because it is undefined at $x = 0$, the curve does not pass through the origin. The right branch descends from $+\infty$ toward $y = 1$, while the left branch descends from $y = -1$ toward $-\infty$. In contrast to $\tanh x$, it diverges near the origin and converges to $\pm 1$ far away.

Relationships with other functions

By definition $\coth x = \dfrac{1}{\tanh x}$, the reciprocal of $\tanh x$. It also satisfies the identity $\coth^2 x - \operatorname{csch}^2 x = 1$.

Applications

$\coth x$ appears frequently in physics. In statistical mechanics it forms part of the Langevin function $L(x) = \coth x - \dfrac{1}{x}$, which describes the magnetization of a paramagnet, and it arises in formulas related to Planck's law of blackbody radiation and Einstein's model of specific heat.