Difference between revisions of "ArcTania"

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m (Text replacement - "\$([^\$]+)\$" to "\\(\1\\)")
 
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[[File:ArcTaniaMap.png|right|350px|thumb| $f=\mathrm{ArcTania}(x\!+\!{\rm i} y)$ in the $x,y$ plane with levels
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[[File:ArcTaniaMap.png|right|350px|thumb| \(f=\mathrm{ArcTania}(x\!+\!{\rm i} y)\) in the \(x,y\) plane with levels
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$u\!=\!\Re(f)\!=\! \mathrm {const} ~$ and
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\(u\!=\!\Re(f)\!=\! \mathrm {const} ~\) and
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$v\!=\!\Im(f)\!=\! \mathrm {const} ~$]]
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\(v\!=\!\Im(f)\!=\! \mathrm {const} ~\)]]
 
[[ArcTania]] is [[elementary function,
  
[[ArcTania]] is [[elementary function,
   
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$\mathrm{ArcTania}(z)=z+\ln(z)−1$
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\(\mathrm{ArcTania}(z)=z+\ln(z)−1\)
   
 
[[Complex map]] of [[ArcTania]] is shown in figure at right.
  
[[Complex map]] of [[ArcTania]] is shown in figure at right.
   
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[[ArcTania]] is important, because its inverse function $\mathrm{Tania}=\mathrm{ArcTania}^{-1}$ , is est, the [[Tania function]],
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[[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
  
appears in the [[Laser science]] as solution of the equation of evolution of intensity of light in the idealised saturable amplifier
 
<ref>
  
<ref>
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</ref>:
  
</ref>:
   
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$ \displaystyle \mathrm{Tania}^{\prime}(z)=\frac{\mathrm{Tania}(z)}{1+\mathrm{Tania}(z)} $
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\( \displaystyle \mathrm{Tania}^{\prime}(z)=\frac{\mathrm{Tania}(z)}{1+\mathrm{Tania}(z)} \)
   
 
==References==
  
==References==

Latest revision as of 18:43, 30 July 2019

\(f=\mathrm{ArcTania}(x\!+\!{\rm i} y)\) 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

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