User:KlappCK

Hello Wikipedia! This user is a computational physicist.

Theorem statement
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} \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). $$