Difference between revisions of "Arcsinh"
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| − | The inverse hyperbolic sine function $\mathrm{arcsinh | + | __NOTOC__ |
| − | + | The inverse hyperbolic sine function $\mathrm{arcsinh}$ is function is the [[inverse function]] of the [[sinh|hyperbolic sine]] function. It may be defined by | |
| − | $$\mathrm{arcsinh}(z)=\log\left(z+\sqrt{1+z^2}\right) | + | $$\mathrm{arcsinh}(z)=\log \left(z + \sqrt{1+z^2} \right),$$ |
| − | + | where $\log$ denotes the [[logarithm]]. | |
<div align="center"> | <div align="center"> | ||
<gallery> | <gallery> | ||
File:Arcsinhplot.png|Plot of $\mathrm{arcsinh}$ on $[-10,10]$. | File:Arcsinhplot.png|Plot of $\mathrm{arcsinh}$ on $[-10,10]$. | ||
| − | File: | + | File:Complexarcsinhplot.png|[[Domain coloring]] of of $\mathrm{arcsinh}$. |
</gallery> | </gallery> | ||
</div> | </div> | ||
=Properties= | =Properties= | ||
| − | + | [[Derivative of arcsinh]]<br /> | |
| + | [[Antiderivative of arcsinh]]<br /> | ||
=See Also= | =See Also= | ||
| Line 22: | Line 23: | ||
[http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_86.htm Abramowitz&Stegun] | [http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_86.htm Abramowitz&Stegun] | ||
| − | + | {{:Inverse hyperbolic trigonometric functions footer}} | |
[[Category:SpecialFunction]] | [[Category:SpecialFunction]] | ||
Latest revision as of 23:28, 11 December 2016
The inverse hyperbolic sine function $\mathrm{arcsinh}$ is function is the inverse function of the hyperbolic sine function. It may be defined by $$\mathrm{arcsinh}(z)=\log \left(z + \sqrt{1+z^2} \right),$$ where $\log$ denotes the logarithm.
Plot of $\mathrm{arcsinh}$ on $[-10,10]$.
Domain coloring of of $\mathrm{arcsinh}$.
Properties
Derivative of arcsinh
Antiderivative of arcsinh
