Tanc function

In mathematics, the Tanc function is defined as^[1]

${\displaystyle \operatorname {Tanc} (z)={\frac {\tan(z)}{z}}}$

Tanc 2D plot

Tanc'(z) 2D plot

Tanc integral 2D plot

Tanc integral 3D plot

Imaginary part in complex plane

• ${\displaystyle \operatorname {Im} \left({\frac {\tan(x+iy)}{x+iy}}\right)}$

Real part in complex plane

• ${\displaystyle \operatorname {Re} \left({\frac {\tan \left(x+iy\right)}{x+iy}}\right)}$

absolute magnitude

• ${\displaystyle \left|{\frac {\tan(x+iy)}{x+iy}}\right|}$

First-order derivative

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

Real part of derivative

• ${\displaystyle -\operatorname {Re} \left(-{\frac {1-(\tan(x+iy))^{2}}{x+iy}}+{\frac {\tan(x+iy)}{(x+iy)^{2}}}\right)}$

Imaginary part of derivative

• ${\displaystyle -\operatorname {Im} \left(-{\frac {1-(\tan(x+iy))^{2}}{x+iy}}+{\frac {\tan(x+iy)}{(x+iy)^{2}}}\right)}$

absolute value of derivative

• ${\displaystyle \left|-{\frac {1-(\tan(x+iy))^{2}}{x+iy}}+{\frac {\tan(x+iy)}{(x+iy)^{2}}}\right|}$

In terms of other special functions

• ${\displaystyle \operatorname {Tanc} (z)={\frac {2\,i{{\rm {KummerM}}\left(1,\,2,\,2\,iz\right)}}{\left(2\,z+\pi \right){{\rm {KummerM}}\left(1,\,2,\,i\left(2\,z+\pi \right)\right)}}}}$ 

• ${\displaystyle \operatorname {Tanc} (z)={\frac {2i\operatorname {HeunB} \left(2,0,0,0,{\sqrt {2}}{\sqrt {iz}}\right)}{(2z+\pi )\operatorname {HeunB} \left(2,0,0,0,{\sqrt {2}}{\sqrt {(i/2)(2z+\pi )}}\right)}}}$ 

• ${\displaystyle \operatorname {Tanc} (z)={\frac {{\rm {WhittakerM}}(0,\,1/2,\,2\,iz)}{{\rm {WhittakerM}}(0,\,1/2,\,i(2z+\pi ))z}}}$ 

Series expansion

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

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

Gallery

Tanc abs complex 3D

Tanc Im complex 3D plot

Tanc Re complex 3D plot

Tanc'(z) Im complex 3D plot

Tanc'(z) Re complex 3D plot

Tanc'(z) abs complex 3D plot

Tanc abs plot

Tanc Im plot

Tanc Re plot

Tanc'(z) Im plot

Tanc'(z) abs plot

Tanc'(z) Re plot

Tanc integral abs plot

Tanc integral Im plot

Tanc integral Re plot

Tanc abs complex 3D plot

Tanc Im complex 3D plot

Tanc Re complex 3D plot

See also

• Sinhc function
• Tanhc function
• Coshc function

References

1. Weisstein, Eric W. "Tanc Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/TancFunction.html