TITLE: Inverting an asymptotic series
QUESTION [5 upvotes]: I have the first few terms of a series of the form, 
$y(x)=\ln(x)+x+a_0+\frac{a_1}{x}+\frac{a_2}{x^2}+\cdots$.
Knowing that the inverse $x(y)$ exists, I am looking for method to write x in terms of y (at least the first few terms of the expansion). Does anybody know how I could achieve this?
Thanks to a mathematician much greater than I, I know that this is certainly possible in the case, the $x$ term is not present in the expansion of $y$ (i.e. $y(x)=\ln(x)+a_0+\frac{a_1}{x}+\frac{a_2}{x^2}+O(\frac{1}{x^3})$). It turns out in this case $x(y)$ can be written as a series expansion in powers of $e^{-y}$. But I can't seem to be able adapt that method to handle this new case. 
Thanks for reading.

REPLY [4 votes]: There is a general technique for doing this, found in expositions dealing with transseries.  One example is my own...  
Transseries for beginners, Real Analysis Exchange 35 (2010) 253--310  
see Problem 4.2.