Lagrange inversion theorem: Difference between revisions

Revision as of 07:31, 9 April 2023

Revision as of 07:44, 20 September 2023

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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 larger {{mvar<math>|ln(z)-1|<{4+\pi^2}}</math> (thoughapproximately not<math>2.58\ldots for\cdot all {10^{mvar|z}-6} < z < 2.869\ldots \cdot 10^6</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>