Transcendental equations involving more than 2 terms

Question

I now how to solve transcendental equations involving only two terms like:

$xe^x=k$

$x=W(k)$

Where W(x) is the Lambert's Omega function.

But how can I solve (for $x$) a more general case? Like:

$xe^x-xe=k$

With $k$ being nonzero.

I mean an exact result, involving well-known functions and not simply an approximation.

Answer 1

Burniston and Siewert built a solution for the equation:

$$ze^z=a(z+b)$$

through an integral representation.

== References ==

[68] C. E. Siewert and E. E. Burniston, "Solutions of the Equation $ze^z=a(z+b)$," Journal of Mathematical Analysis and Applications, 46 (1974) 329-337.

http://www4.ncsu.edu/~ces/pdfversions/68.pdf