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

W1

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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

W2

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It says that it is true? How?!!

Also, another identity:

W4

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

W5

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W3

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