Editing Lagrange inversion theorem (section)

===Lambert ''W'' function===

{{main|Lambert W function}}
The Lambert {{mvar|W}} function is the function <math>W(z)</math> that is implicitly defined by the equation

:<math> W(z) e^{W(z)} = z.</math>

We may use the theorem to compute the [[Taylor series]] of <math>W(z)</math> at <math>z=0.</math> We take <math>f(w) = we^w</math> and <math>a = 0.</math> Recognizing that

:<math>\frac{d^n}{dx^n} e^{\alpha x} = \alpha^n e^{\alpha x},</math>

this gives

:<math>\begin{align}
   W(z) &= \sum_{n=1}^{\infty} \left[\lim_{w \to 0} \frac{d^{n-1}}{dw^{n-1}} e^{-nw} \right] \frac{z^n}{n!} \\
   {} &= \sum_{n=1}^{\infty} (-n)^{n-1} \frac{z^n}{n!} \\
   {} &= z-z^2+\frac{3}{2}z^3-\frac{8}{3}z^4+O(z^5).
 \end{align}</math>

The [[radius of convergence]] of this series is <math>e^{-1}</math> (giving the [[principal branch]] of the Lambert function).

A series that converges for <math>|\ln(z)-1|<\sqrt{{4+\pi^2}}</math> (approximately <math>0.0655 < z < 112.63</math>) can also be derived by series inversion.  The function <math>f(z) = W(e^z) - 1</math> satisfies the equation

:<math>1 + f(z) + \ln (1 + f(z)) = z.</math>

Then <math>z + \ln (1 + z)</math> can be expanded into a power series and inverted.<ref>{{cite conference |url=https://dl.acm.org/doi/pdf/10.1145/258726.258783 |title=A sequence of series for the Lambert W function |last1=Corless |first1=Robert M. |last2=Jeffrey |first2= David J.|author-link2=|last3=Knuth|first3=Donald E.|author-link3=Donald E. Knuth|date=July 1997 |book-title=Proceedings of the 1997 international symposium on Symbolic and algebraic computation |pages=197&ndash;204|doi=10.1145/258726.258783 |url-access=subscription }}</ref>  This gives a series for <math>f(z+1) = W(e^{z+1})-1\text{:}</math>

:<math>W(e^{1+z}) = 1 + \frac{z}{2} + \frac{z^2}{16} - \frac{z^3}{192} - \frac{z^4}{3072} + \frac{13 z^5}{61440} -  O(z^6).</math>

<math>W(x)</math> can be computed by substituting <math>\ln x - 1</math> for {{mvar|z}} in the above series. For example, substituting {{math|−1}} for {{mvar|z}} gives the value of <math>W(1) \approx 0.567143.</math>