From Wikipedia, the free encyclopedia
In mathematics, the Tanc function is defined as[1]
![{\displaystyle \operatorname {Tanc} (z)={\frac {\tan(z)}{z}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5de4aa754b088af2cd8d320e81fb913d352eb9f)
- Imaginary part in complex plane
![\operatorname {Im}\left({\frac {\tan(x+iy)}{x+iy}}\right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/75c8d40715a635029c88135e3a27d289893078e5)
- Real part in complex plane
![\operatorname {Re}\left({\frac {\tan \left(x+iy\right)}{x+iy}}\right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/279499cd7a8556c7aeaf7499c578e52b84df24c5)
- absolute magnitude
![\left|{\frac {\tan(x+iy)}{x+iy}}\right|](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fdac3cf091f3537d25f49c3e8a1d1aa3e9679bb)
- First-order derivative
![{\frac {1-(\tan(z))^{2}}{z}}-{\frac {\tan(z)}{z^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc94dc2ddef0d16c5524f79da0c08162f1237129)
- Real part of derivative
![{\displaystyle -\operatorname {Re} \left(-{\frac {1-(\tan(x+iy))^{2}}{x+iy}}+{\frac {\tan(x+iy)}{(x+iy)^{2}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f9615194d830fac7a2e2848b17d2dd4aafd0a78)
- Imaginary part of derivative
![{\displaystyle -\operatorname {Im} \left(-{\frac {1-(\tan(x+iy))^{2}}{x+iy}}+{\frac {\tan(x+iy)}{(x+iy)^{2}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b46f64f8b7e488d2dfcb350e682871147ab06fb)
- absolute value of derivative
![\left|-{\frac {1-(\tan(x+iy))^{2}}{x+iy}}+{\frac {\tan(x+iy)}{(x+iy)^{2}}}\right|](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b7b61917755482dbeac3f8f8fe494300c719b8)
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)}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fbb70186fcbce46de0f3ffd3eabf27317c7a1cf)
![\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)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/376cffa73b5b219c4ff043f864541054be417ffc)
![{\displaystyle \operatorname {Tanc} (z)={\frac {{\rm {WhittakerM}}(0,\,1/2,\,2\,iz)}{{\rm {WhittakerM}}(0,\,1/2,\,i(2z+\pi ))z}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b94b608c8403c809682ef9c1101e65d49c6f84a)
Series expansion[]
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e96dbb334bcd7f7221467f9c84ac76c3034765f2)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/698cbe678244dab8c4163d082300e26a2b0c2cc0)
Padé approximation[]
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/743c3ec63a90297347a498cd798e00dcf7754c27)
Gallery[]
See also[]
References[]