Residue of inverse function using Lagrange inversion

Question

Let $v(z)$ be a infinite power series of terms $z^k$ where $k>0$ with coefficients $v_k$ and a analytic function except at z=0 and $V(z)$ be the inverse of $v(z)$,then show that $Res_0(V(z)^{-k})=k\times v_k$ at $z=0$ using Lagrange inversion ,could you also extend for any other arbitrary singularity $z_0$ rather than $z=0$.

Answer 1

Suggestion. It is conceptually cleaner for computational manipulations to represent the analytic map's input and output variables by distinct letters. Write $w= f(z)$ and the inverse relation as $z= g(w)$. Then use the integral representation of the residue $$Res[g(w)^{-k},w=0] =\frac{1}{2\pi i} \oint _{w\in C} \frac{ dw}{g^k(w)}$$

This can be evaluated by changing variables so that $g(w)=z$ , then using the substitution $ dw = f'(z) dz$ and expanding that derivative $f'(z)$ in its Taylor expansion in powers of $z$, etc.

P.S. One technical detail: note that the small contour $C$ about $w=0$ is transformed bijectively into some new small contour $C'$ about $z=0$ that has the same orientation as that of $C$.