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I have an equation that can be very likely solved with the Lambert function but looks a bit messy:

$exp(x)-\frac{x^2}{a}+\frac{x}{b}-1=0$, and $a>0$, $b\geq1$ (if constraints help).

Any idea how to get x out of this in the closed form?

Cheers, p

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  • $\begingroup$ Consider Taylor series expansion of $e^x$ say to 3 terms around zero, you could get a quadratic equation that may be useful as an approximate root. $\endgroup$ – NoChance Oct 4 '19 at 19:25
  • $\begingroup$ Approximation is easy, though, I need the exact analytical solution :) $\endgroup$ – peterkey Oct 4 '19 at 19:29
  • $\begingroup$ It helps to specify the domain of $x$, you could replace $e^x$ by a single fraction as in math.stackexchange.com/questions/71357/approximation-of-e-x but an closed form is probably not possible. $\endgroup$ – NoChance Oct 4 '19 at 19:33
  • $\begingroup$ $$x\in [0;1] $$ $\endgroup$ – peterkey Oct 4 '19 at 19:42
  • $\begingroup$ I assume that the approximation in the provided like is excellent. Plot it against the $e^x$ and see how good it is. $\endgroup$ – NoChance Oct 4 '19 at 19:48
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Unless factorization can be done, this requires a generalization of the Lambert W function. See Taylor series for generalized Lambert W function. In essence you can get a series expansion for $x$ in terms of the other variables using Lagrange inversion theorem, but it is likely messy.

Aside from that, not much more can be done to the problem other than numerical computation.

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