simplification of $W(x\cdot e^{a+x})$

Question

Is it possible to simplify $W(x\cdot e^{a+x})$?

Because $W(x\cdot e^{x})=x$

So I was wondering if it was possible to simplify this expression.

Answer 1

I'd have to say no, this is not simplify-able.

You can however attempt to obtain an expansion via logarithms:

$$W(x)=\ln(x)-\ln(\ln(x))+\mathcal O(\ln(\ln(\ln(x))))$$

So,

$$W(xe^{a+x})=a+x+\ln(x)-\ln(a+x+\ln(x))+\mathcal O(\ln(\ln(x)))$$

as $x\to\infty$.