Editing Lambert W function (section)

=== Integer and complex powers ===
Integer powers of {{math|''W''<sub>0</sub>}} also admit simple [[Taylor series|Taylor]] (or [[Laurent series|Laurent]]) series expansions at zero:
: <math>
W_0(x)^2 = \sum_{n=2}^\infty \frac{-2\left(-n\right)^{n-3}}{(n - 2)!} x^n = x^2 - 2x^3 + 4x^4 - \tfrac{25}{3}x^5 + 18x^6 - \cdots.
</math>

More generally, for {{math|''r'' ∈ '''Z'''}}, the [[Lagrange inversion theorem|Lagrange inversion formula]] gives
: <math>
W_0(x)^r = \sum_{n=r}^\infty \frac{-r\left(-n\right)^{n - r - 1}}{(n - r)!} x^n,
</math>
which is, in general, a Laurent series of order {{mvar|r}}. Equivalently, the latter can be written in the form of a Taylor expansion of powers of {{math|''W''<sub>0</sub>(''x'') / ''x''}}:
: <math>
\left(\frac{W_0(x)}{x}\right)^r = e^{-r W_0(x)} = \sum_{n=0}^\infty \frac{r\left(n + r\right)^{n - 1}}{n!} \left(-x\right)^n,
</math>
which holds for any {{math|''r'' ∈ '''C'''}} and {{math|{{abs|''x''}} < {{sfrac|1|''e''}}}}.