Lagrange Inversion Theorem

Let $z$ be defined as a function of $w$ in terms of a parameter $\alpha$ by

$$z=w+\alpha\phi(z).$$

Then any function of $z$ can be expressed as a Power Series in $\alpha$ which converges for sufficiently small $\alpha$ and has the form

$F(z)=F(w)+{\alpha\over 1}\phi(w)F'(w)+{\alpha^2\over 1\cdot 2}{\partial\over\partial w}\{[\phi(w)]^2F'(w)\}$
$+\ldots+{\alpha^{n+1}\over(n+1)!}{\partial^n\over\partial w^n}\{[\phi(w)]^{n+1}F'(w)\}+\ldots.$

References

Goursat, E. Functions of a Complex Variable, Vol. 2, Pt. 1. New York: Dover, 1959.

Moulton, F. R. An Introduction to Celestial Mechanics, 2nd rev. ed. New York: Dover, p. 161, 1970.

Williamson, B. ``Remainder in Lagrange's Series.'' §119 in An Elementary Treatise on the Differential Calculus, 9th ed. London: Longmans, pp. 158-159, 1895.