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I'm not sure if there is any way to simplify $W(xa^x)$. It's pretty clear that $a=e$ simplifies to $x$ or $W_k(xe^x)$, but any other value of $a$, other than trivial values like $a=0,1$, don't seem simplifyable.

Here is a graph you may want to view. It appears to be asymptotic to a linear function is the most I can observe.

From the graph, I observed

$$W(xa^x)\sim\ln(a)x$$

Simply Beautiful Art
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1 Answers1

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This isn't an answer, though I'd really like to see one:

Well, $xa^x=xe^{x \ln(a)}$.

Consequently, if you define: $y=xe^{x \ln(a)}$, it is well defined that:

$x \ln(a)=W(y\ln(a))$

And I'm not sure if that is any more helpful. I'll definitely research a bit more though.

KR136
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  • Thanks for the effort. I did manage to get what you did in my own attempt, but then it hit me that it wasn't actually helpful, at least in any way I could see. :( – Simply Beautiful Art Apr 06 '16 at 19:05