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Inverse of $y=xe^x$

Is there a closed form solution to the equation: $a e^a$ = 1 ?

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marked as duplicate by Qiaochu Yuan Sep 7 '12 at 5:21

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

See – joriki Sep 6 '12 at 22:58
up vote 4 down vote accepted

In general, the solution to the equation $x=a e^a$ for $a$ is given by the Lambert W function, so in this case the solution is just $W(1)$, the so-called Omega Constant. $W(x)$ is not an elementary function, and can not be written in terms of more elementary functions. Whether or not the $W$ function counts as closed-form is up to you, as closed form means different things to different people.

As the Mathworld page above doesn't include any further information, it probably is not known if $W(1)$ itself has a closed form expression in terms of simpler functions. However, since negative results of this type are typically difficult to prove and there's no reason to expect such an expression, I'd strongly bet not even though I'd be surprised at such a proof. For reference, $W(1) \approx 0.56714$.

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It should also be noted that for $-\frac1e<x<0$, there are two values of $a$ so that $x=ae^a$. Therefore, there are two branches of Lambert W for negative arguments. – robjohn Sep 6 '12 at 23:14
@robjohn I meant to restrict to $x>0$ in my answer, but thanks for pointing this out nonetheless. Luckily for $x=1$ there is no need to worry about branch cuts. – Logan Maingi Sep 6 '12 at 23:21
And there are even more than 2 branches if you allow complex numbers... – GEdgar Sep 7 '12 at 0:41

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