Use of the Lagrange–Bürmann formula, which gives the Taylor series expansion of the inverse function of an analytic function.
The Lagrange inversion theorem expresses the Taylor series of analytic function’s inverse or its composition with another function $g(x)$. The general statement of theorem is:
Let $y=f(x)$ and $y_0=f(x_0)$ where $f’(x_0)\ne0$, then
$$g(x)=g(x_0)+\sum_{k=1}^\infty\frac{(y-y_0)^k}{k!}\left\{\frac{d^{k-1}}{dx^{k-1}}\left[g’(x)\left(\frac{x-x_0}{f(x)-y_0}\right)^k\right]\right\}_{x=x_0}$$
