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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

Luke Poeppel
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peterkey
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  • 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. – NoChance Oct 04 '19 at 19:25
  • Approximation is easy, though, I need the exact analytical solution :) – peterkey Oct 04 '19 at 19:29
  • It helps to specify the domain of $x$, you could replace $e^x$ by a single fraction as in https://math.stackexchange.com/questions/71357/approximation-of-e-x but an closed form is probably not possible. – NoChance Oct 04 '19 at 19:33
  • $$x\in [0;1] $$ – peterkey Oct 04 '19 at 19:42
  • I assume that the approximation in the provided like is excellent. Plot it against the $e^x$ and see how good it is. – NoChance Oct 04 '19 at 19:48
  • Yes, unless we can find an analytical solution, 3d order polynomial of Taylor around 0 is more than enough for an approximation. – peterkey Oct 05 '19 at 08:52

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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.

Simply Beautiful Art
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