# ArcTania

Jump to: navigation, search
$$f=\mathrm{ArcTania}(x\!+\!{\rm i} y)$$ 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.