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.