I would appreciate if someone can help me answer the following questions.
Although I read several papers and documents on the Lambert W function, I could not assess on what set is this function (or at least its principle branch) analytic (or holomorphic)? Hence, on what set can one apply the identity theorem on it? I know that the principal branch of the function admits a convergent Taylor series at $0$ with a positive radius of convergence $1/{\rm e}$ given by $$ W_0(z)=\sum\limits_{n=0}^\infty \frac{(-n)^{n-1}}{n!}z^n $$ Hence, $W_0(z)$ is analytic at $0$. However, is it also analytic elsewhere in the complex plane?
According to Wikipedia, "The function defined by this series can be extended to a holomorphic function defined on all complex numbers with a branch cut along the interval $(−\infty, −1/\text{e}]$; this holomorphic function defines the principal branch of the Lambert W function". I don't quite understand this statement. How can the extended function of this set be defined on all complex numbers away from the branch cut, if it only converges for $-\frac{1}{\text{e}}<z<\frac{1}{\text{e}}$ and diverges for all other $z$.
From my understanding, a function is said to be analytic on an open set $D$, if the function converges to its Taylor series in a neighborhood of every point in the set $D$. If the Taylor series of the function $W_0(z)$ at an arbitrary $z_0$ is not known to have a closed-form, does this mean that this function is not analytic at $z_0$? or could it be analytic without a known Taylor series expansion for arbitrary $z_0$? And if the Taylor series around an arbitrary point $z_0$ is not known in closed-form, how can one obtain the radius of convergence of the series? Does the radius of convergence play any role in applying the identity theorem?
In How to derive the Lambert W function series expansion?, there is an example showing how to write the Taylor series expansion of the Lambert W function around $\text{e}$. However, it is not clear to me if this applies to an arbitrary point $z_0$ (oher than $0$ and $\text{e}$) and how can one obtain the radius of converge of this non-closed form series and whether can one claim that the function is analytic at $z_0$?
If we extend the function to the complex domain, it is known that the lambert W function has the derivative $\frac{\text{d}W}{\text{d}z}=\frac{W(z)}{z+\text{e}^{W(z)}}=\frac{W(z)}{z(W(z)+1)}$ for $z\neq \{0,-1/\text{e}\}$. If we differentiate this infinite number of times, and the derivative exists at $z_0$, then the function is infinitely differentiable $z_0$. In this case, it only remains to prove that the function is equal its own Taylor series at a neighborhood of every point of its domain (or some open set) for the function to be holomorphic, right? Can this be shown for arbitrary $z_0$? And hence for some open set?
I tried to apply the Lagrange inversion theorem to get the Taylor series of $W_0(z)$ at arbitrary $z_0$, but I could not converge to a closed-form.
In Short, in my problem, i need to use the identity theorem on the Lambert W function. However, i need to check first on what set this theorem applies. In other words, on what set is the Lambert W function analytic?
Any help is appreciated. Thanks a lot in advance.
