Sinhc function

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

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

The sinhc function is the hyperbolic analogue of the sinc function, defined by $\sin x/x$. It is a solution of the following differential equation: $$w(z)z-2\frac{d}{dz}w(z)-z\frac{d^2}{d{z}^2}w(z)=0$$

Sinhc 2D plot

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

Sinhc'(z) 2D plot

File:Sinhc integral 2D plot.png

Sinhc integral 2D plot

Properties

The first-order derivative is given by

${\mathrm{sinhc}}^{\prime }(z)=\frac{\cosh (z)}{z}-\frac{\sinh (z)}{z^2}$

The Taylor series expansion is$$\sum _{i=0}^{\mathrm{\infty }}\frac{z^{2i}}{(2i+1)!}.$$The Padé approximant is$$\mathrm{sinhc}\left(z\right)=(1+\frac{53272705}{360869676}{z}^2+\frac{38518909}{7217393520}{z}^4+\frac{269197963}{3940696861920}{z}^6+\frac{4585922449}{15605159573203200}{z}^8){(1-\frac{2290747}{120289892}{z}^2+\frac{1281433}{7217393520}{z}^4-\frac{560401}{562956694560}{z}^6+\frac{1029037}{346781323848960}{z}^8)}^{-1}$$

In terms of other special functions

• $\mathrm{sinhc}(z)=\frac{\mathrm{K}\mathrm{u}\mathrm{m}\mathrm{m}\mathrm{e}\mathrm{r}\mathrm{M}(1,2,2z)}{e^z}$, where $\mathrm{K}\mathrm{u}\mathrm{m}\mathrm{m}\mathrm{e}\mathrm{r}\mathrm{M}(a,b,z)$ is Kummer's confluent hypergeometric function.
• $\mathrm{sinhc}(z)=\frac{\mathrm{HeunB}(2,0,0,0,\sqrt{2}\sqrt{z})}{e^z}$, where $\mathrm{H}\mathrm{e}\mathrm{u}\mathrm{n}\mathrm{B}(q,\alpha ,\gamma ,\delta ,ϵ,z)$ is the biconfluent Heun function.
• $\mathrm{sinhc}(z)=1/2\frac{\mathrm{W}\mathrm{h}\mathrm{i}\mathrm{t}\mathrm{t}\mathrm{a}\mathrm{k}\mathrm{e}\mathrm{r}\mathrm{M}(0,1/2,2z)}{z}$, 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:Sinhc abs complex 3D plot.png Sinhc abs complex 3D

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

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

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

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

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

File:Sinhc abs plot.JPG Sinhc abs plot

File:Sinhc Im plot.JPG Sinhc Im plot

File:Sinhc Re plot.JPG Sinhc Re plot

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

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

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

See also

• Tanc function
• Tanhc function
• Sinhc integral
• Coshc function

References

This article "Sinhc function" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Sinhc 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)

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

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