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:<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=197–204|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>
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