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Given function $y = e^x$, the inverse is $x = \ln{y}$. Is it possible to find the inverse of $y = x*e^x$? If this is not possible mathematically, are tools like programs or numerical methods available to do it? I want to put the result, that is, x as a function of y, in an integral. Thank you.

umdas
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    https://en.wikipedia.org/wiki/Lambert_W_function – user388557 Oct 26 '19 at 22:31
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    Since the inverse is not elementary, you might want to see this as well: https://en.wikipedia.org/wiki/Integral_of_inverse_functions – Milten Oct 26 '19 at 22:43
  • @PeldePinda: I think there's enough to say about using Lambert's function to solve the inverse here that you could craft an Answer for har. – hardmath Oct 26 '19 at 22:46
  • It is possible mathematically, of course. Well, that depends on what you mean by find and mathematically, anyway. – Allawonder Oct 27 '19 at 06:29
  • Is my conclusion below correct? I am interested in finding x as a function of y to place in an integral. According to Wikipedia, I can use the following: $x = W_0(y) = y - y^2 + \frac{3}{2}y^3 - \frac{8}{3}y^4$ and so on. The domain of y is zero to one inclusive. – umdas Oct 27 '19 at 19:18
  • Well, no, not for this domain. The series converges for a domain between the absolute value of $1/e$, about .368 – umdas Oct 28 '19 at 10:47

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