Lambert functions, solutions to $W e^W=z$, possess a multi-branched structure due to the exponential term. Each branch, denoted $W_k(z)$, corresponds to a different solution, separated by branch cuts in the complex plane. These cuts, emanating from branch points, prevent ambiguity in the function's value. The principal branch, $W_0(z)$, is typically considered the primary solution. Understanding this branch structure is crucial for correctly evaluating and applying Lambert functions, particularly in areas like complex analysis and differential equations. Visualizing the Riemann surface helps conceptualize this multi-valued nature.
The classical Lambert $W$ function is defined as the multivalued inverse of
$$ W e^W=z $$
It appears in many branches of mathematics and physics and has well-studied asymptotic expansions and branch structures.
The $r$-Lambert function (or the (1, 1)-Lambert function) generalizes it to the equation
$$ W_r e^{a W_r}=z $$
This allows for additional parameters to be introduced into the function's structure. Generalizing to ( $n , m$ )-type Lambert Functions A function of type $(n, m)$ should satisfy an equation of the form:
$$ P(W) e^{Q(W)}=z $$
where $P(W)$ is a polynomial of degree $n$ and $Q(W)$ is a polynomial of degree $m$. This generalization introduces a rich structure of branches, asymptotics, and singularities.
Here, we consider an equation of the form:
$$ \left(W^2+a W\right) e^{b W}=z $$
This function introduces transcendental behavior akin to the classical Lambert function but with additional polynomial nonlinearity.
$$ \frac{d}{d W}\left(\left(W^2+a W\right) e^{b W}\right)=0 $$
This leads to an implicit equation for critical points.
$$ W \approx \frac{1}{b} \log z+O(1) $$
with corrections depending on $a, b$.