I am trying to solve the following Equation for $p$:
$$p=1-e^{\frac{n}{\Omega}-\frac{S}{p\Omega}}$$
My initial idea was to use the Lambert $W$ function, but a colleague told me that this Equation does not have a solution in terms of $W$ because
the Lambert $W$ function does not "like" addition.
If I omit the "$1-$" part, I get the solution
$$p=e^{\frac{n}{\Omega}-\frac{S}{p\Omega}} \iff p=\frac{S}{\Omega W\left(-\frac{Se^{-\frac{n}{\Omega}}}{\Omega}\right)}$$
However, I do not understand why the "$1-$" part in $p=1-e^{\frac{n}{\Omega}-\frac{S}{p\Omega}}$ is so problematic.
Can someone please explain why I cannot use the Lambert $W$ function to solve this Equation for $p$?
Best Answer
Changing notations, you want to solve for $p$ the equation $$p=1-a e^{-\frac b p}$$ Let $p=\frac 1 x$ to make $$e^{-bx}=\frac {1-x} {a x}$$ and the solution is given in terms of the generalized Lambert function (have a look at equation $(4)$).