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I simplified an expression to the term :

1= x- e^(0.616+0.326*x)

using wolfram alpha I get a solution that contains a function called Wn, that I have never heard of. Can someone maybe shed some light onto this, or tell me where to read up to get closer to it. The solution should be 1 or more positive numbers.

Stephan Ketterer
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Welcome to the world of Lambert function ! Your equation being $$1= x- e^{0.616+0.326\,x}$$ rewrite it as $$1=x-e^{0.616} e^{0.326\,x}$$ or making it more general $$1=x-a e^{bx}$$ Following the steps given in the linked page, you should arrive to $$x=1-\frac{W\left(-a\, b\, e^b\right)}{b}$$ where $W(t)$ is Lambert function.

The problem is that, in the real domain, there is no solution to this equation if $t <-\frac 1e$ which is the case with your numbers.

So, the solutions are complex, corresponding to the $W_{-n}$ branches of Lambert function (as Wolfram Alpha told).

Claude Leibovici
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