y = \dfrac{\sinh x}{x}

Graph of the Function $y = \dfrac{\sinh x}{x}$

$y = \dfrac{\sinh x}{x}$ is the hyperbolic sine divided by $x$ . It looks just like the sinc function $\dfrac{\sin x}{x}$ , and yet it behaves entirely differently. Where sinc oscillates and decays, this one does not oscillate at all and simply keeps growing.

Domain and limit

The denominator vanishes at $x = 0$ , so the domain is $x \neq 0$ . But $\sinh x \approx x$ near the origin, so the following limit exists.

\lim_{x \to 0}\frac{\sinh x}{x} = 1

Setting $f(0) = 1$ makes the function continuous on the whole line, and that is the usual definition. It is a removable singularity, and on the graph only that single point is a hole.

Symmetry and range

Both $\sinh$ and $x$ are odd, so their quotient is even and the graph is symmetric about the $y$ -axis. Expanding in a series gives the following.

\frac{\sinh x}{x} = 1 + \frac{x^2}{6} + \frac{x^4}{120} + \cdots = \sum_{n=0}^{\infty}\frac{x^{2n}}{(2n+1)!}

Every term after the first is positive, so $y \geq 1$ and the range is everything from $1$ upward. There are no zeros.

Comparison with sinc

The series for the sinc function has coefficients of the same magnitude, differing only in the alternation of signs.

\frac{\sin x}{x} = 1 - \frac{x^2}{6} + \frac{x^4}{120} - \cdots

That difference of signs alone makes one oscillate and decay to $0$ while the other increases monotonically and diverges.

Item	$\dfrac{\sin x}{x}$	$\dfrac{\sinh x}{x}$
Limit at $x = 0$	$1$	$1$
Range	$\approx -0.217 \leq y \leq 1$	$y \geq 1$
Zeros	$x = n\pi$ with $n \neq 0$	none
Far out	oscillates toward $0$	diverges exponentially
At the origin	maximum	minimum

The relationship between $\sin$ and $\sinh$ carries over unchanged, and putting the two graphs side by side makes the difference plain.

Monotonicity and behavior far out

Examining the derivative shows it is positive for $x > 0$ , so the right half increases monotonically. The function is even, so the left half decreases, and the origin is the minimum.

$x$	$\dfrac{\sinh x}{x}$
$0$	$1$
$1$	$\approx 1.1752$
$2$	$\approx 1.8134$
$3$	$\approx 3.3393$

For large $x$ we have $\sinh x \approx \dfrac{e^{x}}{2}$ , so $y \approx \dfrac{e^{x}}{2x}$ : it diverges at the rate of an exponential divided by $x$ . There is no horizontal asymptote.

Relation to the uniform distribution

Taking the mean of $e^{tx}$ over the uniform distribution on the interval $[-1, 1]$ produces this function¹.

\begin{align*} \frac{1}{2}\int_{-1}^{1}e^{tx}\,dx &= \frac{e^{t} - e^{-t}}{2t} \\ &= \frac{\sinh t}{t} \end{align*}

In other words the function is the moment-generating function of the uniform distribution, a form that appears of its own accord whenever that distribution is handled in probability.

Applications

In the language of spherical Bessel functions, $\dfrac{\sin x}{x}$ is $j_0(x)$ and $\dfrac{\sinh x}{x}$ is the modified spherical Bessel function $i_0(x)$ ². The difference between the two corresponds to the difference between the wave equation, whose solutions oscillate, and the diffusion equation or the equation of an electrostatic field, whose solutions grow or decay exponentially.

Moment-generating function, Wikipedia
Bessel function, Wikipedia

Graph of the Function y=sinh⁡xxy = \dfrac{\sinh x}{x}y=xsinhx​