WrightOmega

WrightOmega is holomorphic funciton, 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" superfunciton 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://www.orcca.on.ca/TechReports/TechReports/2000/TR-00-12.pdf Robert M. Corless, David J. Jeffrey. On the Wright $\omega$ function.