Coshc function

The cardinal hyperbolic cosine function coshc(z) plotted in the complex plane from -2-2i to 2+2i

The cardinal hyperbolic cosine function coshc(z) plotted in the complex plane from -2-2i to 2+2i

In mathematics, the coshc function appears frequently in papers about optical scattering, Heisenberg spacetime and hyperbolic geometry. For $z\ne 0$, it is defined as $$\mathrm{coshc}(z)=\frac{\cosh (z)}{z}$$

It is a solution of the following differential equation: $$w(z)z-2\frac{d}{dz}w(z)-z\frac{d^2}{d{z}^2}w(z)=0$$

File:Coshc 2D plot.png

Coshc 2D plot

File:Coshc'(z) 2D plot.png

Coshc'(z) 2D plot

Properties

The first-order derivative is given by

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

The Taylor series expansion is$$\mathrm{coshc}z\approx (z^{-1}+\frac{1}{2}z+\frac{1}{24}{z}^3+\frac{1}{720}{z}^5+\frac{1}{40320}{z}^7+\frac{1}{3628800}{z}^9+\frac{1}{479001600}{z}^{11}+\frac{1}{87178291200}{z}^{13}+O(z^{15}))$$

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

In terms of other special functions

• $\mathrm{coshc}(z)=\frac{(iz+1/2\pi )\mathrm{M}(1,2,i\pi -2z)}{e^{(i/2)\pi -z}z}$, where $\mathrm{M}(a,b,z)$ is Kummer's confluent hypergeometric function.
• $\mathrm{coshc}(z)=\frac{1}{2}\frac{(2iz+\pi )\mathrm{HeunB}(2,0,0,0,\sqrt{2}\sqrt{1/2i\pi -z})}{e^{1/2i\pi -z}z}$, where $\mathrm{H}\mathrm{e}\mathrm{u}\mathrm{n}\mathrm{B}(q,\alpha ,\gamma ,\delta ,ϵ,z)$ is the biconfluent Heun function.
• $\mathrm{coshc}(z)=\frac{-i(2iz+\pi )\mathrm{W}\mathrm{h}\mathrm{i}\mathrm{t}\mathrm{t}\mathrm{a}\mathrm{k}\mathrm{e}\mathrm{r}\mathrm{M}(0,1/2,i\pi -2z)}{(4iz+2\pi )z}$, where $\mathrm{W}\mathrm{h}\mathrm{i}\mathrm{t}\mathrm{t}\mathrm{a}\mathrm{k}\mathrm{e}\mathrm{r}\mathrm{M}(a,b,z)$ is a Whittaker function.

Gallery

File:Coshc abs complex 3D plot.png Coshc abs complex 3D

File:Coshc Im complex 3D plot.png Coshc Im complex 3D plot

File:Coshc Re complex 3D plot.png Coshc Re complex 3D plot

File:Coshc'(z) Im complex 3D plot.png Coshc'(z) Im complex 3D plot

File:Coshc'(z) Re complex 3D plot.png Coshc'(z) Re complex 3D plot

File:Coshc'(z) abs complex 3D plot.png Coshc'(z) abs complex 3D plot

File:Coshc'(x) abs density plot.JPG Coshc'(x) abs density plot

File:Coshc'(x) Im density plot.JPG Coshc'(x) Im density plot

File:Coshc'(x) Re density plot.JPG Coshc'(x) Re density plot

See also

• Tanc function
• Tanhc function
• Sinhc function

References

This article "Coshc function" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Coshc function. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

📰 Article(s) of the same category(ies)

• Tanc function
• Tanhc function
• Sinhc function
• Approximation Theory and Special Functions

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