what is exactly analytic continuation of the product log function

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When I solve in wolfram equation like this $xe^x=z$ they give me the solution $x=W_n(z)$

I know about $x=W_0(z) $ and $x=W_{1}(z)$ but for $n$ I searched in the internet but I didn't find anything can give me an expression about $x=W_n(z)$

Please can anyone help me and give me an explain $x=W_n(z)$ and give me an example how we can find $x=W_n(t)$ $t\in C$ or $t\in R$

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1 Answer

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All that, and much more, is here:

Corless, R.M.; Gonnet, G.H.; Hare, D.E.G.; Jeffrey, D.J.; and Knuth, D.E. "On the Lambert W Function." Advances in Computational Mathematics, Vol. 5, (1996): 329-359.

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