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
$$\mathrm{arctanh}(z) = \dfrac{\log(1+z)}{2} - \dfrac{\log(1-x)}{2},$$
+
$$\mathrm{arctanh}(z) = \dfrac{\log(1+z)}{2} - \dfrac{\log(1-z)}{2},$$
 
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:Complex ArcTanh.jpg|[[Domain coloring]] of [[analytic continuation]] of $\mathrm{arctanh}$.
+
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.

Properties

Derivative of arctanh
Antiderivative of arctanh
Derivative of Legendre chi

See also

Arctan
Tanh
Tangent

Inverse hyperbolic trigonometric functions