Lambert W of 2πi

Question

$$2πi = 2πi, $$

$$2πi = 2πi \cdot e^{2πi}$$ Where $$e^{2πi} =1$$

$$W(2πi) = W(2πi \cdot e^{2πi}),$$ $$W(2πi) = 2πi$$ Where $$ W(xe^x) = x$$

When I check whether the last statement is valid in WolframAlpha, it tellls me that it is not. What have I done wrong?

Answer 1

Thanks to Achille Hui.

Apparently the Lambert W function has a seperate branch for the complex numbers. Hence $W_1(2πi) =2πi $, while $W_0(2πi) \neq2πi $ WolframAlpha shows LambertW[1,2*Pi*i] to be $2πi$.