Lagrange inversion theorem: Difference between revisions

Revision as of 23:49, 4 March 2025

Revision as of 10:28, 18 March 2025

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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; 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/389556414_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>

For <math>f(0)=0</math>, <math>f'(0)\neq0</math>, analytic around 0, continuous on <math>[0,a)</math>, where <math>a</math> is a real constant so that <math>f(t)>0</math> for all <math>t\in(0,a]</math> (it can be moddified to work for <math>f(0)\neq0</math>).

Nevertheless, is not as easy to compute as it requires the [[Analytic continuation]] of the integral for all real <math>s</math>, but it's quite powerful if applicable.

Returning to Lagrange's inversion theorem;

If the assertions about analyticity are omitted, the formula is also valid for [[formal power series]] and can be generalized in various ways: It can be formulated for functions of several variables; it can be extended to provide a ready formula for {{math|''F''(''g''(''z''))}} for any analytic function {{mvar|F}}; and it can be generalized to the case <math>f'(a)=0,</math> where the inverse {{mvar|g}} is a multivalued function.