Tanhc function

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

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

In mathematics, the tanhc function is defined for $z\ne 0$ as $$\mathrm{tanhc}(z)=\frac{\tanh (z)}{z}$$The tanhc function is the hyperbolic analogue of the tanc function.

File:Tanhc 2D plot.png

Tanhc 2D plot

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

Tanhc'(z) 2D plot

File:Tanhc integral 2D plot.png

Tanhc integral 2D plot

File:Tanhc integral 3D plot.png

Tanhc integral 3D plot

Properties

The first-order derivative is given by

$\frac{{\mathrm{sech}}^2(z)}{z}-\frac{\tanh (z)}{z^2}$

The Taylor series expansion$$\mathrm{tanhc}z\approx (1-\frac{1}{3}{z}^2+\frac{2}{15}{z}^4-\frac{17}{315}{z}^6+\frac{62}{2835}{z}^8-\frac{1382}{155925}{z}^{10}+\frac{21844}{6081075}{z}^{12}-\frac{929569}{638512875}{z}^{14}+O(z^{16}))$$which leads to the series expansion of the integral as$$\underset{0}{\overset{z}{\int }}\frac{\tanh \left(x\right)}{x}dx=(z-\frac{1}{9}{z}^3+\frac{2}{75}{z}^5-\frac{17}{2205}{z}^7+\frac{62}{25515}{z}^9-\frac{1382}{1715175}{z}^{11}+O\left(z^{13}\right))$$

The Padé approximant is$$\mathrm{tanhc}\left(z\right)=(1+\frac{7}{51}{z}^2+\frac{1}{255}{z}^4+\frac{2}{69615}{z}^6+\frac{1}{34459425}{z}^8){(1+\frac{8}{17}{z}^2+\frac{7}{255}{z}^4+\frac{4}{9945}{z}^6+\frac{1}{765765}{z}^8)}^{-1}$$

In terms of other special functions

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

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

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

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

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

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

File:Tanhc abs plot.JPG Tanhc abs plot

File:Tanhc Im plot.JPG Tanhc Im plot

File:Tanhc Re plot.JPG Tanhc Re plot

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

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

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

File:Tanhc integral abs 3D plot.png Tanhc integral abs 3D plot

File:Tanhc integral Im 3D plot.png Tanhc integral Im 3D plot

File:Tanhc integral Re 3D plot.png Tanhc integral Re 3D plot

File:Tanhc integral abs density plot.JPG Tanhc integral abs density plot

File:Tanhc integral Im density plot.JPG Tanhc integral Im density plot

File:Tanhc integral Re density plot.JPG Tanhc integral Re density plot

See also

• Sinhc function
• Tanc function
• Coshc function

References

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

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

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