The relation between Wright Omega Function and Lambert W Function

lambert-w

I have a logarithmic equation: $u = A\left(\ln(u)+1\right)$. I used MATLAB's symbolic toolbox to solve this equation for u. 

syms u  A
solve(u == A*log(u+1),u)
ans = - A*wrightOmega(- log(-A) - 1/A) - 1

I looked up the Wright Omega function and wikipedia says it is defined by the Lambert W function. I was expecting the solution to the equation would have a Lambert W term. So, I am wondering if there is a relation that I can apply to write the solution with Lambert W instead of Wright Omega.

Best Answer

Taken directly from Wikipedia, the Wright Omega function, $\omega(z)$, is defined with respect to the Lambert W function, $W_k(z)$, for complex argument, $z$, as:

$$\omega(z) = W_{\big \lceil \frac{\mathrm{Im}(z) - \pi}{2 \pi} \big \rceil}(e^z) \Longleftrightarrow W_k(z) = \omega(\ln(z) + 2 \pi i k)$$

So, assuming that your parameter $A>0$ and is real, your function can be rewritten and simplified as:

$$-A W_{-1}\left(\frac{-e^{-1/A}}{A}\right)-1$$

or -A*lambertw(-1,-exp(-1/A)/A)-1 in Matlab.

Ultimately, the lesser-known Wright Omega function is a simpler function than the Lambert W function. You can evaluate it numerically in Matlab with my wrightOmegaq, which is available on Github and on the MathWorks File Exchange.

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