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\(f=\mathrm{ArcTania}(x\!+\!{\rm i} y)\) in the \(x,y\) plane with levels \(u\!=\!\Re(f)\!=\! \mathrm {const} ~\) and \(v\!=\!\Im(f)\!=\! \mathrm {const} ~\)

ArcTania is [[elementary function,


Complex map of ArcTania is shown in figure at right.

ArcTania is important, because its inverse function \(\mathrm{Tania}=\mathrm{ArcTania}^{-1}\) , is est, the Tania function, appears in the Laser science as solution of the equation of evolution of intensity of light in the idealised saturable amplifier [1]:

\( \displaystyle \mathrm{Tania}^{\prime}(z)=\frac{\mathrm{Tania}(z)}{1+\mathrm{Tania}(z)} \)


  1. D.Kouznetsov. Superfunctions for amplifiers. Optical Review, July 2013, Volume 20, Issue 4, pp 321-326.


Doya function, Inverse function, Tania