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

File:Tanc integral 2D plot.png

Tanc integral 2D 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

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

• ${\displaystyle \operatorname {Tanc} (z)=2{\frac {\operatorname {HeunB} (2,0,0,0,{\sqrt {2}}{\sqrt {z}})}{(2iz+\pi )\operatorname {HeunB} (2,0,0,0,{\sqrt {2}}{\sqrt {1/2\,i\pi -z}})e^{2\,z-1/2\,i\pi }}}}$ 

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

Series expansion

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

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

References

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