Sinhc function

In mathematics, the sinhc function appears frequently in papers about optical scattering,^[1] Heisenberg spacetime^[2] and hyperbolic geometry.^[3]^{[better source needed]} For $z \neq 0$ , it is defined as^[4]^[5] $sinhc (z) = \frac{\sinh (z)}{z}$

The sinhc function is the hyperbolic analogue of the sinc function, defined by $\sin x / x$ . It is a solution of the following differential equation: $w (z) z - 2 \frac{d}{d z} w (z) - z \frac{d^{2}}{d z^{2}} w (z) = 0$

Properties

The first-order derivative is given by

\frac{\cosh (z)}{z} - \frac{\sinh (z)}{z^{2}}

The Taylor series expansion is $\sum_{i = 0}^{\infty} \frac{z^{2 i}}{(2 i + 1)!} .$ The Padé approximant is $sinhc (z) = (1 + \frac{53272705}{360869676} z^{2} + \frac{38518909}{7217393520} z^{4} + \frac{269197963}{3940696861920} z^{6} + \frac{4585922449}{15605159573203200} z^{8}) {(1 - \frac{2290747}{120289892} z^{2} + \frac{1281433}{7217393520} z^{4} - \frac{560401}{562956694560} z^{6} + \frac{1029037}{346781323848960} z^{8})}^{- 1}$

In terms of other special functions

$sinhc (z) = \frac{K u m m e r M (1, 2, 2 z)}{e^{z}}$ , where $K u m m e r M (a, b, z)$ is Kummer's confluent hypergeometric function.
$sinhc (z) = \frac{HeunB (2, 0, 0, 0, \sqrt{2} \sqrt{z})}{e^{z}}$ , where $H e u n B (q, α, γ, δ, ϵ, z)$ is the biconfluent Heun function.
$sinhc (z) = 1 / 2 \frac{W h i t t a k e r M (0, 1 / 2, 2 z)}{z}$ , where $W h i t t a k e r M (a, b, z)$ is a Whittaker function.