Coshc function

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

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

It is a solution of the following differential equation:

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

Coshc 2D plot

Coshc'(z) 2D plot

Imaginary part in complex plane

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

Real part in complex plane

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

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

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

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

• $\operatorname {Coshc} (z)={\frac {-i(2\,iz+\pi ){{\rm {\mathbf {W} hittakerM}}(0,\,1/2,\,i\pi -2z)}}{(4iz+2\pi )z}}$

Series expansion

$\operatorname {Coshc} z\approx \left(z^{-1}+{\frac {1}{2}}z+{\frac {1}{24}}z^{3}+{\frac {1}{720}}z^{5}+{\frac {1}{40320}}z^{7}+{\frac {1}{3628800}}z^{9}+{\frac {1}{479001600}}z^{11}+{\frac {1}{87178291200}}z^{13}+O(z^{15})\right)$

Padé approximation

$\operatorname {Coshc} \left(z\right)={\frac {23594700729600+11275015752000\,{z}^{2}+727718024880\,{z}^{4}+13853547000\,{z}^{6}+80737373\,{z}^{8}}{147173\,{z}^{9}-39328920\,{z}^{7}+5772800880\,{z}^{5}-522334612800\,{z}^{3}+23594700729600\,z}}$

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

Coshc'(x) abs density plot

Coshc'(x) Im density plot

Coshc'(x) Re density plot

See also

• Tanc function
• Tanhc function
• Sinhc function