Graph of the Hyperbolic Cosecant $y = \operatorname{csch} x$

The hyperbolic cosecant function $y = \operatorname{csch} x$ is defined as the reciprocal of the hyperbolic sine¹.

It is the hyperbolic counterpart of the ordinary cosecant $\csc x$. It pairs with the hyperbolic secant $\operatorname{sech} x = \dfrac{1}{\cosh x}$, but since $\cosh x$ never vanishes while $\sinh x$ does at the origin, the two behave quite differently.

Domain and range

• The domain is $x \neq 0$
• The range is $y \neq 0$
• It decreases monotonically on each branch
• It is an odd function

Where $\operatorname{sech}$ was defined on the whole line and bounded within $(0, 1]$, this function has a domain split in two and values that grow without bound.

Symmetry and sign

Since $\sinh$ is odd, its reciprocal is odd too, and the graph is symmetric about the origin. It is positive for $x > 0$ and negative for $x < 0$, so the curve occupies the first and third quadrants.

Monotonicity

The derivative is $\dfrac{d}{dx}\operatorname{csch} x = -\operatorname{csch} x \coth x$. For $x > 0$ both $\operatorname{csch} x$ and $\coth x$ are positive, so the derivative is negative; for $x < 0$ both are negative, so their product is positive and the derivative is again negative. The function therefore decreases on both branches and has no extrema.

Concavity and asymptotes

The second derivative is $\dfrac{\cosh^{2}x + 1}{\sinh^{3}x}$. The numerator is always positive, so the sign follows $\sinh^{3}x$, which is the sign of $x$. The curve is concave up for $x > 0$ and concave down for $x < 0$, with no inflection point.

The $y$-axis is a vertical asymptote and the $x$-axis a horizontal one.

Two faces

Near the origin $\sinh x \approx x$, so the curve is all but indistinguishable from an inverse proportion, while far out it falls exponentially to $0$.

Indeed $\operatorname{csch}(0.001) \approx 999.9998$, almost exactly the $\dfrac{1}{x} = 1000$ that an inverse proportion would give. The function wears two faces: an inverse proportion near the origin and an exponential decay far away.

Identities and integral

Corresponding to $\csc^{2}\theta - \cot^{2}\theta = 1$ for trigonometric functions, the following holds.

There is also the half-argument relation $\coth x - \operatorname{csch} x = \tanh\dfrac{x}{2}$. The antiderivative corresponds directly to the trigonometric case.

1. Hyperbolic functions, Wikipedia