I am interested in the complex-analysis version of deriving Lagrange's inversion theorem:
If $y=f(x)$ with $f(a)=b$ and f'(a)\neq 0, then $x(y)=a+\sum_{n=1}^{\infty} \left(\lim_{x\to a}\frac{d^{n-1}}{dx^{n-1}}\left(\frac{x-a}{f(x)-b}\right)^n \frac{(y-b)^n}{n!}\right).$
The derivative expression immediately suggests some type of residue calculation but I'm not able to derive it or find any reference on line that goes over it.
Can someone here help me with this?