Difference between revisions of "Arctanh"
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The inverse hyperbolic tangent function $\mathrm{arctanh}$ is the [[inverse function]] of the [[tanh|hyperbolic tangent]] function. It may be defined by | The inverse hyperbolic tangent function $\mathrm{arctanh}$ is the [[inverse function]] of the [[tanh|hyperbolic tangent]] function. It may be defined by | ||
| − | $$\mathrm{arctanh}(z) = \dfrac{\log(1+z)}{2} - \dfrac{\log(1- | + | $$\mathrm{arctanh}(z) = \dfrac{\log(1+z)}{2} - \dfrac{\log(1-z)}{2},$$ |
where $\log$ denotes the [[logarithm]]. | where $\log$ denotes the [[logarithm]]. | ||
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<gallery> | <gallery> | ||
File:Arctanhplot.png|Plot of $\mathrm{arctanh}$ on $(-1,1)$. | File:Arctanhplot.png|Plot of $\mathrm{arctanh}$ on $(-1,1)$. | ||
| − | File: | + | File:Complexarctanhplot.png|[[Domain coloring]] of $\mathrm{arctanh}$. |
</gallery> | </gallery> | ||
</div> | </div> | ||
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=Properties= | =Properties= | ||
[[Derivative of arctanh]] <br /> | [[Derivative of arctanh]] <br /> | ||
| + | [[Antiderivative of arctanh]]<br /> | ||
[[Derivative of Legendre chi]] <br /> | [[Derivative of Legendre chi]] <br /> | ||
Latest revision as of 23:47, 11 December 2016
The inverse hyperbolic tangent function $\mathrm{arctanh}$ is the inverse function of the hyperbolic tangent function. It may be defined by $$\mathrm{arctanh}(z) = \dfrac{\log(1+z)}{2} - \dfrac{\log(1-z)}{2},$$ where $\log$ denotes the logarithm.
Plot of $\mathrm{arctanh}$ on $(-1,1)$.
Domain coloring of $\mathrm{arctanh}$.
Properties
Derivative of arctanh
Antiderivative of arctanh
Derivative of Legendre chi
