Editing Lambert W function (section)

== Asymptotic expansions ==
The [[Taylor series]] of {{math|''W''<sub>0</sub>}} around 0 can be found using the [[Lagrange inversion theorem]] and is given by
: <math>W_0(x)=\sum_{n=1}^\infty \frac{(-n)^{n-1}}{n!}x^n =x-x^2+\tfrac{3}{2}x^3-\tfrac{16}{6}x^4+\tfrac{125}{24}x^5-\cdots.</math>

The [[radius of convergence]] is {{math|{{sfrac|1|''e''}}}}, as may be seen by the [[ratio test]]. 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 (mathematics)|interval]] {{open-closed|−∞, −{{sfrac|1|''e''}}}}; this holomorphic function defines the [[principal branch]] of the Lambert {{mvar|W}} function.

For large values of {{mvar|x}}, {{math|''W''<sub>0</sub>}} is asymptotic to
: <math>\begin{align}
W_0(x) &= L_1 - L_2 + \frac{L_2}{L_1} + \frac{L_2\left(-2 + L_2\right)}{2L_1^2} + \frac{L_2\left(6 - 9L_2 + 2L_2^2\right)}{6L_1^3} + \frac{L_2\left(-12 + 36L_2 - 22L_2^2 + 3L_2^3\right)}{12L_1^4} + \cdots \\[5pt]
&= L_1 - L_2 + \sum_{l=0}^\infty \sum_{m=1}^\infty \frac{(-1)^l \left[ \begin{smallmatrix} l + m \\ l + 1 \end{smallmatrix} \right]}{m!} L_1^{-l-m} L_2^m,
\end{align}</math>
where {{math|1=''L''<sub>1</sub> = ln ''x''}}, {{math|1=''L''<sub>2</sub> = ln ln ''x''}}, and {{math|<big><big>[</big></big>{{su|p=''l'' + ''m''|b=''l'' + 1|a=c}}<big><big>]</big></big>}} is a non-negative [[Stirling numbers of the first kind|Stirling number of the first kind]].<ref name = "Corless" /> Keeping only the first two terms of the expansion,
: <math>W_0(x) = \ln x - \ln \ln x + \mathcal{o}(1).</math>

The other real branch, {{math|''W''<sub>−1</sub>}}, defined in the interval {{closed-open|−{{sfrac|1|''e''}}, 0}}, has an approximation of the same form as {{mvar|x}} approaches zero, with in this case {{math|1=''L''<sub>1</sub> = ln(−''x'')}} and {{math|1=''L''<sub>2</sub> = ln(−ln(−''x''))}}.<ref name = "Corless" />

=== 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''}}}}.