Lambert W called Product Log, More than 2 Branches

This is really two questions.

Why is the Lambert W function alternatively called the product log? I have not found any reference to why it is called that, only that it occasionally is.
On the Wikipedia page for the Lambert W function, it states that the function has two branches, W0W_0 and W−1W_{-1}. But on a paper on the Wright Omega function, the authors state that there are infinite branches:

Lambert W satisfies W(z)exp(W(z))=zW(z)exp(W(z))=z, and has an infinite number of branches, denoted Wk(z)W_k(z), for k∈Zk\in \Bbb Z.

Which is correct?




WW has two real branches, but there are more complex-valued branches.
– arkeet
2 days ago



I think the name productLog comes from the fact that log(z)\log(z) satisfies elog(z)=ze^{\log(z)} = z while W(z)W(z) satisfies W(z)eW(z)=zW(z)e^{W(z)}= z
– user1952009
2 days ago


1 Answer


The Lambert W is the inverse of

f(z)=zezf(z) = ze^z

Since the inverse of f(z)=ezf(z) = e^z is called logarithm, it makes sense to call the inverse of the product zezze^z product logarithm.