Difference between revisions of "Arcsinh"
From specialfunctionswiki
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| + | The $\mathrm{arcsinh}$ function is the inverse function of the [[sinh|hyperbolic sine]] function defined by | ||
| + | $$\mathrm{arcsinh}(z)=\log\left(z+\sqrt{1+z^2}\right).$$ | ||
| + | |||
[[File:Complex ArcSinh.jpg|500px]] | [[File:Complex ArcSinh.jpg|500px]] | ||
| + | |||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$\dfrac{d}{dz} \mathrm{arcsinh}(z) = \dfrac{1}{\sqrt{1+z^2}}.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
<center>{{:Inverse hyperbolic trigonometric functions footer}}</center> | <center>{{:Inverse hyperbolic trigonometric functions footer}}</center> | ||
Revision as of 05:44, 16 May 2015
The $\mathrm{arcsinh}$ function is the inverse function of the hyperbolic sine function defined by $$\mathrm{arcsinh}(z)=\log\left(z+\sqrt{1+z^2}\right).$$
Properties
Theorem: The following formula holds: $$\dfrac{d}{dz} \mathrm{arcsinh}(z) = \dfrac{1}{\sqrt{1+z^2}}.$$
Proof: █
