Lagrange inversion theorem: Difference between revisions

Revision as of 20:06, 2 March 2025

Revision as of 23:49, 4 March 2025

Line 15:

The theorem further states that this series has a non-zero radius of convergence, i.e., <math>g(z)</math> represents an analytic function of {{mvar|z}} in a [[neighbourhood (mathematics)|neighbourhood]] of <math>z= f(a).</math> This is also called '''reversion of series'''.

There is an alternative representation derived directly from this theorem that works for non-finite polynomial functions; In a paper released in 2025: "A Closed form Solution to Kepler's Equation"<ref>Santos, Patricio Asis (2025). "A Closed-Form Solution to Kepler's Equation" https://www.researchgate.net/publication/389253839_A_Closed389556414_A_Closed-Form_Solution_to_Kepler's_Equation</ref> by P. Santos, the terms of the following expression are derived at the end of the third page:

<math>g(z)=\sum_{n=1}^{\infty}\frac{z^{n}}{n}\lim_{s\to n}(n-s)\int_{0}^{a}f(t)^{-s}dt</math>