Identities of Lambert W Function

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

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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

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It says that it is true? How?!!

Also, another identity:

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

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Ian Fowler

May 9Edited: May 9

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

Omair Siddique

May 9

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

Ian Fowler

May 9

@Omair Siddique You're quite welcome. As long as you allow the output to be Complex.

blackpenredpen

math for fun

Identities of Lambert W Function