Coshc function

In mathematics, the coshc 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] $\operatorname {coshc} (z)={\frac {\cosh(z)}{z}}$

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

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

The Taylor series expansion is $\operatorname {coshc} z\approx \left(z^{-1}+{\frac {1}{2}}z+{\frac {1}{24}}z^{3}+{\frac {1}{720}}z^{5}+{\frac {1}{40320}}z^{7}+{\frac {1}{3628800}}z^{9}+{\frac {1}{479001600}}z^{11}+{\frac {1}{87178291200}}z^{13}+O(z^{15})\right)$

The Padé approximant is $\operatorname {Coshc} \left(z\right)={\frac {23594700729600+11275015752000\,{z}^{2}+727718024880\,{z}^{4}+13853547000\,{z}^{6}+80737373\,{z}^{8}}{147173\,{z}^{9}-39328920\,{z}^{7}+5772800880\,{z}^{5}-522334612800\,{z}^{3}+23594700729600\,z}}$

In terms of other special functions

$\operatorname {coshc} (z)={\frac {(iz+1/2\,\pi ){\rm {M}}(1,2,i\pi -2z)}{e^{(i/2)\pi -z}z}}$ , where ${\rm {M}}(a,b,z)$ is Kummer's confluent hypergeometric function.
$\operatorname {coshc} (z)={\frac {1}{2}}\,{\frac {(2\,iz+\pi )\operatorname {HeunB} \left(2,0,0,0,{\sqrt {2}}{\sqrt {1/2\,i\pi -z}}\right)}{e^{1/2\,i\pi -z}z}}$ , where ${\rm {HeunB}}(q,\alpha ,\gamma ,\delta ,\epsilon ,z)$ is the biconfluent Heun function.
$\operatorname {coshc} (z)={\frac {-i(2\,iz+\pi ){{\rm {\mathbf {W} hittakerM}}(0,\,1/2,\,i\pi -2z)}}{(4iz+2\pi )z}}$ , where ${\rm {WhittakerM}}(a,b,z)$ is a Whittaker function.