Lagrange Expansion

Let $y=f(x)$ and $y_0=f(x_0)$ where $f'(x_0)\not=0$, then

$$x=x_0+\sum_{k=1}^\infty {(y-y_0)^k\over k!}\left\{{{d^{k-1}\... ...dx^{k-1}}\left[{x-x_0\over f(x)-y_0}\right]^k}\right\}_{x=x_0}$$

$$g(x)=g(x_0)+\sum_{k=1}^\infty {(y-y_0)^k\over k!}\left\{{{d^... ...\left({x-x_0\over f(x)-y_0}\right)^k}\right]}\right\}_{x=x_0}.$$

See also Maclaurin Series, Taylor Series

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 14, 1972.