Tanc function

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

${\displaystyle {\frac {\sec ^{2}(z)}{z}}-{\frac {\tan(z)}{z^{2}}}}$

The Taylor series expansion is

$${\displaystyle \operatorname {tanc} z\approx \left(1+{\frac {1}{3}}z^{2}+{\frac {2}{15}}z^{4}+{\frac {17}{315}}z^{6}+{\frac {62}{2835}}z^{8}+{\frac {1382}{155925}}z^{10}+{\frac {21844}{6081075}}z^{12}+{\frac {929569}{638512875}}z^{14}+O(z^{16})\right)}$$

which leads to the series expansion of the integral as

$${\displaystyle \int _{0}^{z}{\frac {\tan(x)}{x}}\,dx=\left(z+{\frac {1}{9}}z^{3}+{\frac {2}{75}}z^{5}+{\frac {17}{2205}}z^{7}+{\frac {62}{25515}}z^{9}+{\frac {1382}{1715175}}z^{11}+{\frac {21844}{79053975}}z^{13}+{\frac {929569}{9577693125}}z^{15}+O(z^{17})\right)}$$

The Padé approximant is

$${\displaystyle \operatorname {tanc} \left(z\right)=\left(1-{\frac {7}{51}}\,{z}^{2}+{\frac {1}{255}}\,{z}^{4}-{\frac {2}{69615}}\,{z}^{6}+{\frac {1}{34459425}}\,{z}^{8}\right)\left(1-{\frac {8}{17}}\,{z}^{2}+{\frac {7}{255}}\,{z}^{4}-{\frac {4}{9945}}\,{z}^{6}+{\frac {1}{765765}}\,{z}^{8}\right)^{-1}}$$

• Sinhc function
• Tanhc function
• Coshc function