Tanhc function

Tanhc function is defined as^[1]

${\displaystyle tanhc(z)={\frac {\tanh \left(z\right)}{z}}}$

Tanhc 2D plot

Tanhc'(z) 2D plot

Tanhc integral 2D plot

• ${\displaystyle tanhc(z)={\frac {\tanh \left(z\right)}{z}}}$

Imaginary part in complex plane

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

Real part in complex plane

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

absolute magnitude

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

First order derivative

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

Real part of derivative

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

Imaginary part of derivative

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

absolute value of derivative

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

In terms of other special functions

• ${\displaystyle tanhc(z)=2\,{\frac {{\rm {KummerM}}\left(1,\,2,\,2\,z\right)}{\left(2\,iz+\pi \right){{\rm {KummerM}}\left(1,\,2,\,i\pi -2\,z\right)}{{\rm {e}}^{2\,z-1/2\,i\pi }}}}}$ 
• ${\displaystyle tanhc(z)=2\,{\frac {{\it {HeunB}}\left(2,0,0,0,{\sqrt {2}}{\sqrt {z}}\right)}{\left(2\,iz+\pi \right){\it {HeunB}}\left(2,0,0,0,{\sqrt {2}}{\sqrt {1/2\,i\pi -z}}\right){{\rm {e}}^{2\,z-1/2\,i\pi }}}}}$ 

Gallery

Tanhc abs complex 3D

Tanhc Im complex 3D plot

Tanhc Re complex 3D plot

Tanhc'(z) Im complex 3D plot

Tanhc'(z) Re complex 3D plot

Tanhc'(z) abs complex 3D plot

Tanhc abs plot

Tanhc Im plot

Tanhc Re plot

Tanhc'(z) Im plot

Tanhc'(z) abs plot

Tanhc'(z) Re plot

References

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