Coshc function

In mathematics, the Coshc function appears frequently in papers about optical scattering,Heisenberg Spacetimeand hyperbolic geometry.It is defined as

$\operatorname {Coshc} (z)={\frac {\cosh(z)}{z}}$

It is a solution of the following differential equation:

$w\left(z\right)z-2\,{\frac {d}{dz}}w\left(z\right)-z{\frac {d^{2}}{d{z}^{2}}}w\left(z\right)=0$

Coshc 2D plot

Coshc'(z) 2D plot

File:Coshc integral 2D plot.png

Coshc integral 2D plot

Imaginary part in complex plane

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

Real part in complex plane

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

absolute magnitude

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

First-order derivative

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

Real part of derivative

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

Imaginary part of derivative

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

absolute value of derivative

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

In terms of other special functions

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

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

• $\operatorname {Coshc} (z)={\frac {-i\left(2\,iz+\pi \right){{\rm {\bf {M}}}\left(0,\,1/2,\,i\pi -2\,z\right)}}{\left(4\,iz+2\,\pi \right)z}}$

Series expansion

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

Gallery

Coshc abs complex 3D

Coshc Im complex 3D plot

Coshc Re complex 3D plot

Coshc'(z) Im complex 3D plot

Coshc'(z) Re complex 3D plot

Coshc'(z) abs complex 3D plot

File:Coshc abs plot.JPG Coshc abs plot

File:Coshc Im plot.JPG Coshc Im plot

File:Coshc Re plot.JPG Coshc Re plot

File:Coshc'(z) Im plot.JPG Coshc'(z) Im plot

File:Coshc'(z) abs plot.JPG Coshc'(z) abs plot

File:Coshc'(z) Re plot.JPG Coshc'(z) Re plot

See also

Tanc function Tanhc function Sinhc function