# Approximate solution for an exponential equation

Trying to solve this question: Probability of ball ownership I got at an expression for the solution, P:

$$\frac{P}{M} = (1 - \frac{1}{M+N})^{N(1-\frac{P}{M})+M}$$

Where M, N are parameters.

The problem is, I got P both in the left side and in the exponent in the right side, and I have no idea how to simplify it.

My goal is to have a simple approximation formula for P, as a function of M and N.

(I already solved it for several values of M, N using numeric methods, but I want a simple formula).

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The Lambert function will be needed for this. On the other hand, your computer will very likely use numerical methods for evaluating the logarithm and the Lambert function anyway, so what's the point? –  Guess who it is. Apr 29 '12 at 17:49
Oh, and: $$p=\frac{m}{n \log \left(1-\frac{1}{m+n}\right)}W\left(n \left(1-\frac{1}{m+n}\right)^{m+n} \log \left(1-\frac{1}{m+n}\right)\right)$$ –  Guess who it is. Apr 29 '12 at 17:51
Great! Since m and n are relatively large integers, this can be approximated as: $$\frac{-m(m+n)}{n} W(\frac{-n}{e(m+n)})$$ Is there a simpler approximation to W, in this specific case? –  Erel Segal-Halevi Apr 29 '12 at 18:23
@ErelSegalHalevi - If $m<<n$, you can use $W(-1/e)=1$. –  nbubis Apr 29 '12 at 18:35
"Is there a simpler approximation to $W$?" - sure, you can take $$W(x)\approx \frac{ex}{1+\left(\frac1{e-1}-\frac1{\sqrt 2}+\frac1{\sqrt{2ex+2}}\right)^{-1}}$$ for instance. (The approximation is due to Serge Winitzki.) –  Guess who it is. Apr 30 '12 at 0:33

## 1 Answer

The Lambert W-function can be used to solve this.

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