Difference between revisions of "Arcsinh"
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| + | File:Arcsinhplot.png|Plot of $\mathrm{arcsinh}$ on $[-10,10]$. | ||
File:Complex ArcSinh.jpg|[[Domain coloring]] of [[analytic continuation]] of $\mathrm{arcsinh}$. | File:Complex ArcSinh.jpg|[[Domain coloring]] of [[analytic continuation]] of $\mathrm{arcsinh}$. | ||
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Revision as of 09:32, 9 November 2015
The $\mathrm{arcsinh}$ function is the inverse function of the hyperbolic sine function defined by $$\mathrm{arcsinh}(z)=\log\left(z+\sqrt{1+z^2}\right).$$
Plot of $\mathrm{arcsinh}$ on $[-10,10]$.
Domain coloring of analytic continuation of $\mathrm{arcsinh}$.
Properties
Theorem: The following formula holds: $$\dfrac{d}{dz} \mathrm{arcsinh}(z) = \dfrac{1}{\sqrt{1+z^2}}.$$
Proof: █
