In mathematics, the tanc function is defined for {\displaystyle z\neq 0} as[1]
{\displaystyle \operatorname {tanc} (z)={\frac {\tan(z)}{z}}}
Properties
The first-order derivative of the tanc function is given by
e {\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}}
In terms of other special functions
{\displaystyle \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 {\displaystyle {\rm {KummerM}}(a,b,z)} is Kummer's confluent hypergeometric function.
{\displaystyle \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 {\displaystyle {\rm {HeunB}}(q,\alpha ,\gamma ,\delta ,\epsilon ,z)} is the biconfluent Heun function.
{\displaystyle \operatorname {tanc} (z)={\frac {{\rm {WhittakerM}}(0,\,1/2,\,2\,iz)}{{\rm {WhittakerM}}(0,\,1/2,\,i(2z+\pi ))z}}}, where {\displaystyle {\rm {WhittakerM}}(a,b,z)} is a Whittaker function.
Gallery
See also
Sinhc function
Tanhc function
Coshc function
References
Weisstein, Eric W. "Tanc Function". mathworld.wolfram.com. Retrieved 2022-11-17.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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