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



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

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


With $k$ being nonzero.

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


Burniston and Siewert built a solution for the equation:


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.



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