Difference between revisions of "WrightOmega"

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[[WrightOmega]] is holomorphic funciton, solution $f$ of equations
 +
[[WrightOmega]] is holomorphic function, solution \(f\) of equations
   
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: $\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (1) ~ ~ ~
 +
: \(\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (1) ~ ~ ~
 
f'(z)= \frac{f(z)}{1+f(z)}
  
f'(z)= \frac{f(z)}{1+f(z)}
  +
\)
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$
  
 
with initial condition
  
with initial condition
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: $\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (2) ~ ~ ~
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: \(\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (2) ~ ~ ~
 
f(1)=1
  
f(1)=1
  +
\)
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$
  
 
<!--
  
<!--
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The contour of integration of equation (1) goes from zero to the imaginary part of $z$ along the imaginary axis and then along the straight line (parallel to the real axis) to the point $z$.
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The contour of integration of equation (1) goes from zero to the imaginary part of \(z\) along the imaginary axis and then along the straight line (parallel to the real axis) to the point \(z\).
 
The contour of integration of equation (1) can be modified without to affect the values of the function, but it should not intersect the cut lines
 
The contour of integration of equation (1) can be modified without to affect the values of the function, but it should not intersect the cut lines
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$ \Re(z)\!<\!-1$, $\Im(z)\!=\!\pm \pi$. In particular, for the real values of the argument, the integration of (1) from 0 to $z$ can be performed along the real axis.
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\( \Re(z)\!<\!-1\), \(\Im(z)\!=\!\pm \pi\). In particular, for the real values of the argument, the integration of (1) from 0 to \(z\) can be performed along the real axis.
 
!-->
  
!-->
 
In vicinity of the real axis, [[WrightOmega]] coincides with the [[Tania function]] of displaced argument.
 
In vicinity of the real axis, [[WrightOmega]] coincides with the [[Tania function]] of displaced argument.
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As the [[Tania function]], as [[WrightOmega]] describe the evolution of intensity of light in an idealized hoomgeneous amplifier with simple model of the gain medium; in the dimension-less form that model gives equation (1).
  
As the [[Tania function]], as [[WrightOmega]] describe the evolution of intensity of light in an idealized hoomgeneous amplifier with simple model of the gain medium; in the dimension-less form that model gives equation (1).
   
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Function [[Filog]]$(a)$, that returns value of the fixed point of logarithm and used in definition of [[tetration]] for complex base $b=\log(a)$, can be expressed through the [[WrightOmega]].
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Function [[Filog]]\((a)\), that returns value of the fixed point of logarithm and used in definition of [[tetration]] for complex base \(b=\log(a)\), can be expressed through the [[WrightOmega]].
   
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: $\displaystyle \mathrm{Filog}(z)= \frac{\mathrm{Tania}\!\big(\ln(z)-1-\mathrm{i}\big)}{-z} = \frac{\mathrm{WrightOmega}\!\big(\ln(z)-\mathrm{i}\big)}{-z}$
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: \(\displaystyle \mathrm{Filog}(z)= \frac{\mathrm{Tania}\!\big(\ln(z)-1-\mathrm{i}\big)}{-z} = \frac{\mathrm{WrightOmega}\!\big(\ln(z)-\mathrm{i}\big)}{-z}\)
   
 
The care about the cut lines of is necessary for the application of the formula above, as the evaluation happens at the edge of the cut line.
  
The care about the cut lines of is necessary for the application of the formula above, as the evaluation happens at the edge of the cut line.
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<ref>
  
<ref>
 
http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183523043
  
http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183523043
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E.M.Wright. Solution of the equation $z\, e^z = a$.
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E.M.Wright. Solution of the equation \(z\, e^z = a\).
 
Bulletin of the Americal Mathematical Society, v. 65, p.89-93 (1959).
  
Bulletin of the Americal Mathematical Society, v. 65, p.89-93 (1959).
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</ref> as solution $f$ of equation
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</ref> as solution \(f\) of equation
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<!--: $f\, \mathrm e^f = z$!-->
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<!--: \(f\, \mathrm e^f = z\)!-->
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: $ f + \ln(f) = z$
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: \( f + \ln(f) = z\)
   
 
In such a way, the inverse of the WrightOmega is elementary function [[zex]]:
  
In such a way, the inverse of the WrightOmega is elementary function [[zex]]:
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: $ \mathrm{ArcWrightOmega}(z)=z\,\exp(z)=\mathrm{zex}(z)$
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: \( \mathrm{ArcWrightOmega}(z)=z\,\exp(z)=\mathrm{zex}(z)\)
   
 
In [[TORI]], historycally, the [[WrightOmega]] is implemented through the [[Tania function]];
  
In [[TORI]], historycally, the [[WrightOmega]] is implemented through the [[Tania function]];
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: $\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (3) ~ ~ ~ \mathrm{WrightOmega}(z)=\mathrm{Tania}(z\!-\!1)$
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: \(\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (3) ~ ~ ~ \mathrm{WrightOmega}(z)=\mathrm{Tania}(z\!-\!1)\)
   
 
Describing and implementing the [[Tania function]], the Editor did not know that the similar function [[WrightOmega]] is already described in literature
  
Describing and implementing the [[Tania function]], the Editor did not know that the similar function [[WrightOmega]] is already described in literature
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http://www.orcca.on.ca/TechReports/TechReports/2000/TR-00-12.pdf
  
http://www.orcca.on.ca/TechReports/TechReports/2000/TR-00-12.pdf
 
Robert M. Corless, David J. Jeffrey.
  
Robert M. Corless, David J. Jeffrey.
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On the Wright $\omega$ function.
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On the Wright \(\omega\) function.
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</ref>. However, neither the efficient algorithms, nor the complex maps for the [[WrightOmega]] are presented in the descriptions cited; so, at least for year 2012, the [[Tania function]] is considered as "principal" superfunciton of the [[Doya function]]. The condition $\mathrm{Tania}(0)=1$ is choosen in analogy with other [[superfunction]]s, where value at zero is choosen as minimal integer that is still larger than the [[fixed point]] used for the construction with [[regular iteration]].
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</ref>. However, neither the efficient algorithms, nor the complex maps for the [[WrightOmega]] are presented in the descriptions cited; so, at least for year 2012, the [[Tania function]] is considered as "principal" superfunction of the [[Doya function]]. The condition \(\mathrm{Tania}(0)=1\) is choosen in analogy with other [[superfunction]]s, where value at zero is choosen as minimal integer that is still larger than the [[fixed point]] used for the construction with [[regular iteration]].
 
For this reason, both functions, [[Tania function]] and [[WrightOmega]] are used in [[TORI]].
  
For this reason, both functions, [[Tania function]] and [[WrightOmega]] are used in [[TORI]].
   
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The WrightOmega is related to the [[LambertW]] function, and in certain sense simpler, as all the branches $\mathrm{LambertW}_k$ of [[LambertW]] function can be expresses through the WrightOmega in the following way:
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The WrightOmega is related to the [[LambertW]] function, and in certain sense simpler, as all the branches \(\mathrm{LambertW}_k\) of [[LambertW]] function can be expresses through the WrightOmega in the following way:
   
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: $\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (4) ~ ~ ~
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: \(\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (4) ~ ~ ~
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\mathrm{LambertW}_k(z)=\mathrm{WrightOmega}(\ln(z)+2\pi \mathrm i k)$
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\mathrm{LambertW}_k(z)=\mathrm{WrightOmega}(\ln(z)+2\pi \mathrm i k)\)
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, at least for integer values of $k$. However, the same relation can be postulated for non–integer values of $k$, extending definition of $\mathrm{LambertW}_k$.
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, at least for integer values of \(k\). However, the same relation can be postulated for non–integer values of \(k\), extending definition of \(\mathrm{LambertW}_k\).
   
 
Both [[WrightOmega]] and [[Tania function]] are [[Superfunction]]s of the [[Doya function]].
  
Both [[WrightOmega]] and [[Tania function]] are [[Superfunction]]s of the [[Doya function]].

Latest revision as of 10:22, 20 July 2020

WrightOmega is holomorphic function, solution \(f\) of equations

\(\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (1) ~ ~ ~ f'(z)= \frac{f(z)}{1+f(z)} \)

with initial condition

\(\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (2) ~ ~ ~ f(1)=1 \)

In vicinity of the real axis, WrightOmega coincides with the Tania function of displaced argument.

Uses of WrightOmega

As the Tania function, as WrightOmega describe the evolution of intensity of light in an idealized hoomgeneous amplifier with simple model of the gain medium; in the dimension-less form that model gives equation (1).

Function Filog\((a)\), that returns value of the fixed point of logarithm and used in definition of tetration for complex base \(b=\log(a)\), can be expressed through the WrightOmega.

\(\displaystyle \mathrm{Filog}(z)= \frac{\mathrm{Tania}\!\big(\ln(z)-1-\mathrm{i}\big)}{-z} = \frac{\mathrm{WrightOmega}\!\big(\ln(z)-\mathrm{i}\big)}{-z}\)

The care about the cut lines of is necessary for the application of the formula above, as the evaluation happens at the edge of the cut line.

History of the function

Originally, properties of the WrightOmega function had been published in 1959 by E.M.Wright [1] as solution \(f\) of equation

\( f + \ln(f) = z\)

In such a way, the inverse of the WrightOmega is elementary function zex:

\( \mathrm{ArcWrightOmega}(z)=z\,\exp(z)=\mathrm{zex}(z)\)

In TORI, historycally, the WrightOmega is implemented through the Tania function;

\(\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (3) ~ ~ ~ \mathrm{WrightOmega}(z)=\mathrm{Tania}(z\!-\!1)\)

Describing and implementing the Tania function, the Editor did not know that the similar function WrightOmega is already described in literature [2][3]. However, neither the efficient algorithms, nor the complex maps for the WrightOmega are presented in the descriptions cited; so, at least for year 2012, the Tania function is considered as "principal" superfunction of the Doya function. The condition \(\mathrm{Tania}(0)=1\) is choosen in analogy with other superfunctions, where value at zero is choosen as minimal integer that is still larger than the fixed point used for the construction with regular iteration. For this reason, both functions, Tania function and WrightOmega are used in TORI.

The WrightOmega is related to the LambertW function, and in certain sense simpler, as all the branches \(\mathrm{LambertW}_k\) of LambertW function can be expresses through the WrightOmega in the following way:

\(\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (4) ~ ~ ~ \mathrm{LambertW}_k(z)=\mathrm{WrightOmega}(\ln(z)+2\pi \mathrm i k)\)

, at least for integer values of \(k\). However, the same relation can be postulated for non–integer values of \(k\), extending definition of \(\mathrm{LambertW}_k\).

Both WrightOmega and Tania function are Superfunctions of the Doya function.

References

  1. ↑ http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183523043 E.M.Wright. Solution of the equation \(z\, e^z = a\). Bulletin of the Americal Mathematical Society, v. 65, p.89-93 (1959).
  2. ↑ http://en.wikipedia.org/wiki/Wright_Omega_function
  3. ↑ http://www.orcca.on.ca/TechReports/TechReports/2000/TR-00-12.pdf Robert M. Corless, David J. Jeffrey. On the Wright \(\omega\) function.

Keywords

Tania function, Doya function, LambertW,

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