ArcTania

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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,

$\mathrm{ArcTania}(z)=z+\ln(z)−1$

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)} $

References

  1. http://link.springer.com/article/10.1007/s10043-013-0058-6
    http://mizugadro.mydns.jp/PAPERS/2013or1.pdf
    http://mizugadro.mydns.jp/PAPERS/2013or2.pdf D.Kouznetsov. Superfunctions for amplifiers. Optical Review, July 2013, Volume 20, Issue 4, pp 321-326.

Keywords

Doya function, Inverse function, Tania