Hi, I was recently viewing a Wikipedia article on Lambert W Function. I was going through the identities and found this identity:

W1PNG

It says that it is true of all x>= than -1/e. I typed this identity in WolframAlpha and inserted a value of -2/e, which is outside the given domain. And here is what I got

W2PNG

It says that it is true? How?!!

Also, another identity:

W4PNG

According to my observation, the given domain should be 0<=a<=e

W5PNG

W3PNG

I forgot to mention that ln(2) -1 + pi(i) is the principal value

Thanks for the explanation. So, it means that input value can be any real number. 😊