# User:KlappCK

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## Theorem statement

Suppose ${\displaystyle y}$ is defined as a function of ${\displaystyle x}$ by an equation of the form

${\displaystyle f(x)=y\,}$

where ${\displaystyle f}$ is analytic at a point ${\displaystyle c}$ and ${\displaystyle f'(c)}$ ≠ 0. Then it is possible to invert or solve the equation for ${\displaystyle x}$:

${\displaystyle x=f^{-1}(y)\,}$

on a neighbourhood of ${\displaystyle f(c)}$, where ${\displaystyle f^{-1}(c)}$ is analytic at the point ${\displaystyle f(c)}$. This is also called reversion of series.

The series expansion of ${\displaystyle f^{-1}}$ is:

${\displaystyle f^{-1}(y)=c+\sum _{n=1}^{\infty }\left(\lim _{x\to c}\left({\frac {(y-f(c))^{n}}{n!}}{\frac {\mathrm {d} ^{\,n-1}}{\mathrm {d} x^{\,n-1}}}\left({\frac {x-c}{f(x)-f(c)}}\right)^{n}\right)\right).}$