# What is the relation between Wright Omega Function and Lambert W Function?

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.

• I've used the MathJax to write the equation but don't know how to format the code in MathJax – anikfaisal Mar 20 '19 at 0:32
• Sorry I misunderstood the content. The convention is to have a code block by intending 4 spaces (necessary for each line). – Lee David Chung Lin Mar 20 '19 at 0:37
• Thanks! I'll keep that in mind – anikfaisal Mar 20 '19 at 1:07

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.