Lambert W(z)在$-e^{-1}$的ramification index为2

hbghlyj

uwo.ca/apmaths/faculty/jeffrey/pdfs/wbranch.pdf

wolframalpha.com/input?i=branch+points+of+W(z)

这次WolframAlpha还列出了ramification index

$z$	ramification index
$0$	logarithmic
$-\frac1e$	$2$
$\tilde\infty$	logarithmic

functions.wolfram.com/ElementaryFunctions/ProductLog/06/ShowAll.html
Expansions at $z=-1/e$
$W(z) \sim-1+O\left(\sqrt{z+\frac{1}{e}}\right)$

我有一个问题：这是怎么推出的？它为什么在$z=-1/e$的ramification index为2？

hbghlyj

plot primepi(x), x/W(x), x/(1+1/e)/W(x/(1+1/e))  from x = 0 to 500