# Residue of inverse function using Lagrange inversion

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 $$z=0$$.

• ? What have you tried? Also, there a few variations on the theme of "Lagrange inversion". Which one are you using? This needs a bit more deth to it, as a question Commented Apr 27 at 12:49
• I have tried Lagrange burmann formula,but I can't prove this residue formula.I am having difficulty figuring it out. Commented Apr 27 at 13:09

## 1 Answer

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 $$C$$.

• What would happen if V(a)=0 . Commented Apr 28 at 18:36