#complex analysis

The inverse function theorem is as follows:

To begin The proof, let C be a circle in the complex plane. We then apply Cauchy's integral formula over the image of C under f to the above function:

Now, the integral over the image of a path under an analytic function equals the integral of the path through a modified integrand. This is essentially pre-applying chain rule to the path integral.

This allows us to rewrite our above expression:

From here we can trivially find the inverse to be the primitive of this function, yielding the formula:

QED

some notes:

We didn't actually need to use the inverse function theorem. As it turns out, this trick can be used for the case f^-1(z)=f^-1(z), yeilding

which, as far as I can tell, is still a further depressed version of the Lagrange inversion theorem. However, there is one strength the formula proven above: it lends itself to a very strong description of nonstandard inverse functions. When interpreted as a line integral, the primitive naturally keeps the structure of the inverse in mind, in a "Riemannian way," as far as I can tell. The latter formula doesn't immediately communicate such a property to me, and the Lagrange inversion theorem technically outputs a power series.

Thanks for reading, hope I didn't waste your time!