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

Sinhc'(z) 2D plot

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

Sinhc abs complex 3D

Sinhc Im complex 3D plot

Sinhc Re complex 3D plot

Sinhc'(z) Im complex 3D plot

Sinhc'(z) Re complex 3D plot

Sinhc'(z) abs complex 3D plot

Sinhc abs plot

Sinhc Im plot

Sinhc Re plot

Sinhc'(z) Im plot

Sinhc'(z) abs plot

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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