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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.

AlienRem
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  • Your problem seems very similiar to:

    http://math.stackexchange.com/questions/1033398/lambert-w-function-with-rational-polynomial/1179641#1179641

    And it involves a generalization of Lambert W.

     – giorgiomugnaini Mar 17 '15 at 16:04
  • In http://math.stackexchange.com/questions/428057/the-positive-root-of-the-transcendental-equation-ln-x-sqrtx-11-0

    a solution of a more general transcendental equation has been found in terms of InverseGammaRegularized[a, s].

     – giorgiomugnaini Mar 18 '15 at 08:01
  • Yes, I know about this result but how can I apply that in my case ? – AlienRem Mar 18 '15 at 14:23

1 Answers1

4

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

giorgiomugnaini
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