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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 $p$?

mto_19
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    Your colleague is probably thinking over the fact that the Lambert is defined as an inverse to a multiplication. It is hard / impossible to isolate an addition of knowns and unknowns as a multiplication of one known and one unknown, so you’ll have trouble applying $W$ successfully. I’m not saying it can’t be done though – FShrike Sep 13 '22 at 07:35

1 Answers1

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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 $(4)$).

Claude Leibovici
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