Tanc Function

In mathematics, the tanc function is defined for $z \neq 0$ as
$\operatorname(z)=\frac$

Properties

The first-order derivative of the tanc function is given by
:$\frac - \frac$
The Taylor series expansion is$\operatorname z \approx \left(1+ \frac z^2 + \frac z^4 + \frac z^6 + \frac z^8 + \frac z^ + \frac z^+ \frac z^ + O(z^ ) \right)$which leads to the series expansion of the integral as$\int _0^z \frac \, dx = \left(z+ \frac z^3 + \frac z^5 + \frac z^7 + \frac z^9+ \frac z^+ \frac z^ + \frac z^+ O (z^) \right)$The Padé approximant is$\operatorname \left( z \right) = \left( 1-\,^ + \,^-\,^+\,^ \right) \left( 1-\,^+\,^-\,^+\,^ \right) ^$

In terms of other special functions

* $\operatorname(z)=$, where $(a,b,z)$ is Kummer's confluent hypergeometric function.
*$\operatorname(z)= \frac$, 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

* Sinhc function
* Tanhc function
* Coshc function

References

Special functions