Do we know of any exact values of the lambert W function like we do for, say, the gamma function at $\frac 12$ ?


Yes , you have for example :

$$ W(e)=1 , W(0)=0 , w(-\frac{\pi}{2})=\pm i\frac{\pi}{2}$$

The last identity comes from the mulitvaluated character of $W$.

  • $\begingroup$ Thanks a lot :) ! $\endgroup$ – Adam Jun 18 '18 at 10:21
  • $\begingroup$ You're welcome. $\endgroup$ – Pagode Jun 18 '18 at 10:26

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