If have a function $f$ given by a series $$ f(z) = \sum_{n,m = 0}^\infty u_{n,m} z^{n + m t} $$ for some $t\in\mathbb{R}^+$.

Is there an straightforward way (something similar to the inversion formula for Taylor series) to derive a similar series for $f^{-1}(z)$?

  • $\begingroup$ Lagrange inversion? en.wikipedia.org/wiki/Lagrange_inversion_theorem $\endgroup$ – Lord Shark the Unknown Mar 22 at 22:08
  • $\begingroup$ I'm not sure how that would work in this case, since it's not exactly a Taylor series $\endgroup$ – Dark Malthorp Mar 22 at 22:10
  • $\begingroup$ What is the inversion formula for Taylor series ? $\endgroup$ – Yves Daoust Mar 22 at 22:11
  • $\begingroup$ With $w:=z^t$, you in fact have a bivariate series $f(z,w)=\sum u_{n,m}z^nw^m$. $\endgroup$ – Yves Daoust Mar 22 at 22:14
  • $\begingroup$ OK. Does that help me invert it? $\endgroup$ – Dark Malthorp Mar 23 at 1:04

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