Tanc function

In mathematics, the Tanc function is defined as

$\operatorname {Tanc} (z)={\frac {\tanh(z)}{z}}$

Imaginary part in complex plane

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

Real part in complex plane

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

absolute magnitude

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

First-order derivative

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

Real part of derivative

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

Imaginary part of derivative

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

absolute value of derivative

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

In terms of other special functions

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

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

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

Series expansion

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