Sinhc function

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

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: $w(z)z-2\,{\frac {d}{dz}}w(z)-z{\frac {d^{2}}{dz^{2}}}w(z)=0$

Properties

The first-order derivative is given by

\operatorname {sinhc} ^{\prime }(z)={\frac {\cosh(z)}{z}}-{\frac {\sinh(z)}{z^{2}}}

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

In terms of other special functions

$\operatorname {sinhc} (z)={\frac {{\rm {KummerM}}(1,\,2,\,2\,z)}{e^{z}}}$ , where ${\rm {KummerM}}(a,b,z)$ is Kummer's confluent hypergeometric function.
$\operatorname {sinhc} (z)={\frac {\operatorname {HeunB} \left(2,0,0,0,{\sqrt {2}}{\sqrt {z}}\right)}{e^{z}}}$ , where ${\rm {HeunB}}(q,\alpha ,\gamma ,\delta ,\epsilon ,z)$ is the biconfluent Heun function.
$\operatorname {sinhc} (z)=1/2\,{\frac {{\rm {WhittakerM}}(0,\,1/2,\,2\,z)}{z}}$ , where ${\rm {WhittakerM}}(a,b,z)$ is a Whittaker function.