1

$$2πi = 2πi, $$

$$2πi = 2πi \cdot e^{2πi}$$ 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?

Anirudh Yamunan Govindarajan
  • 125
  • Does $W$ even have a clear definition over the complex numbers – is $z \longmapsto ze^z$ a bijection? – Aphelli Mar 26 '22 at 07:33
  • 2
    Lambert W is not a single valued function. See: https://en.wikipedia.org/wiki/Lambert_W_function – Kavi Rama Murthy Mar 26 '22 at 07:33
  • 1
    The Lambert W function is multi-valued, its value depend on which branch you pick. On branch $1$ (instead of the default branch $0$?), $W(2\pi i)$ do equal to $2\pi i$. On WA, you can use LambertW[1,2*Pi*i] to compute that value. – achille hui Mar 26 '22 at 08:50
  • Thanks a lot! Could you write this as an answer and also provide a definition of the Lambert W on branch 1? – Anirudh Yamunan Govindarajan Mar 26 '22 at 09:21

1 Answers1

0

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 $2πi$.

Anirudh Yamunan Govindarajan
  • 125