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Theorem statement[edit]

Suppose y is defined as a function of x by an equation of the form

f(x) = y\,

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

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

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

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

  f^{-1}(y) = c
  + \sum_{n=1}^{\infty}
\lim_{x \to c}\left(
{\frac{(y - f(c))^n}{n!}}
\left( \frac{x - c}{f(x) - f(c)} \right)^n\right)