Tanc function

In mathematics, the tanc function is defined for $z\neq 0$ as^[1] $\operatorname {tanc} (z)={\frac {\tan(z)}{z}}$

Properties

The first-order derivative of the tanc function is given by

{\frac {1-(\tan(z))^{2}}{z}}-{\frac {\tan(z)}{z^{2}}}

The Taylor series expansion is $\operatorname {tanc} z\approx \left(1+{\frac {1}{3}}z^{2}+{\frac {2}{15}}z^{4}+{\frac {17}{315}}z^{6}+{\frac {62}{2835}}z^{8}+{\frac {1382}{155925}}z^{10}+{\frac {21844}{6081075}}z^{12}+{\frac {929569}{638512875}}z^{14}+O(z^{16})\right)$ which leads to the series expansion of the integral as $\int _{0}^{z}{\frac {\tan(x)}{x}}\,dx=\left(z+{\frac {1}{9}}z^{3}+{\frac {2}{75}}z^{5}+{\frac {17}{2205}}z^{7}+{\frac {62}{25515}}z^{9}+{\frac {1382}{1715175}}z^{11}+{\frac {21844}{79053975}}z^{13}+{\frac {929569}{9577693125}}z^{15}+O(z^{17})\right)$ The Padé approximant is $\operatorname {tanc} \left(z\right)=\left(1-{\frac {7}{51}}\,{z}^{2}+{\frac {1}{255}}\,{z}^{4}-{\frac {2}{69615}}\,{z}^{6}+{\frac {1}{34459425}}\,{z}^{8}\right)\left(1-{\frac {8}{17}}\,{z}^{2}+{\frac {7}{255}}\,{z}^{4}-{\frac {4}{9945}}\,{z}^{6}+{\frac {1}{765765}}\,{z}^{8}\right)^{-1}$

In terms of other special functions

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