Coshc Function

In mathematics, the coshc function appears frequently in papers about optical scattering, Heisenberg spacetime and hyperbolic geometry. For $z \neq 0$, it is defined as
$\operatorname(z)=\frac$

It is a solution of the following differential equation:
$w( z) z-2\frac w (z) -z \frac w (z) =0$

Properties

The first-order derivative is given by
:$\frac - \frac$
The Taylor series expansion is$\operatorname z \approx \left(z^+\frac z+\frac z^3+\frac z^5+\frac z^7+\frac z^9+\frac z^+\frac z^+O(z^) \right)$

The Padé approximant is$\operatorname \left( z \right) =$

In terms of other special functions

* $\operatorname(z) = \frac$, where $(a,b,z)$ is Kummer's confluent hypergeometric function.
*$\operatorname(z)=\frac\,\frac$, where $(q, \alpha, \gamma, \delta, \epsilon ,z)$ is the biconfluent Heun function.
* $\operatorname(z)= \frac$, where $(a,b,z)$ is a Whittaker function.

Gallery

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See also

* Tanc function
* Tanhc function
* Sinhc function

References

Special functions