Cross-posted from Math Stackexchange.


In an older question to which I provided an [answer](https://math.stackexchange.com/questions/4119317/let-fx-fracx-ln-x1x-evaluate-lim-m-to-f-leftx-0-right-fracx-1/4135364#4135364) it was asked how to compute a particular limit involving the roots of a transcedental function around its extremum. This limit required the evaluation of several terms of a power series for each one of the two roots the function possesses around the minimum, which is reminiscent of a procedure akin to the Lagrange inversion theorem. However, as explicitly stated in the assumptions of the theorem, the derivative of the function has to be non-zero for the theorem to apply (the function has to be locally invertible). That requirement however, did not stop me from deriving a term by term expansion for both functional inverses around the maximum.


The reason why I was surprised is because I couldn't find any references on the subject, and this calculation seems to be a low-hanging fruit of a simple generalization to a well-known theorem.


*My question* is twofold, but an answer to either components will suffice:


1. Assuming that around a point $x\_0$, where $f'(x\_0)=0, f''(x\_0)\neq 0$ and given that $f(x)=\sum\_n a\_n(x-x\_0)^n$, what are the coefficients $b^{\pm}\_{n}$ of the series expansions of the two functional inverses $r^\pm(x)=\sum\_{n=0}^{\infty}b\_n^{\pm}(x-f(x\_0))^{n/2}$, that satisfy $f(r^\pm (x))=x$? They also can be seen as the two solutions to the equation $f(x)=c$ in a neighborhood around $c=f(x\_0)$. Is there a general formula for them (however formal?) Here I define $r^+(x)$ to be the root satisfying $r^+(x)>x\_0$ and the other one such that $r^-(x)<x\_0$.


2. Has this structure been studied in the literature before? Does it come under a certain name? If not, what is it that makes this theorem difficult to establish or uninteresting?


*My work:* Experimenting with a couple of simple functions and their behavior around double zeroes indicates that these coefficients are related by $b\_n^-=(-1)^n b\_n^+$. Also, these series expansions are obviously one-sided: they are defined for $x\in [f(x\_0),R)$ if $f''(x\_0)>0$ and for $x\in (R, f(x\_0)]$ if $f''(x\_0)<0$, for some value of $R$ representing a radius of convergence. I also computed the first few coefficients for arbitrary $f$ with a minimum at $x\_0$


$$b\_0=x\_0,~~ b\_1=\sqrt{\frac{2}{f''(x\_0)}}, ~~ b\_2=-\frac{f'''(x\_0)}{3(f''(x\_0))^2}$$


I also noticed that in a further generalization of the problem, when I demand that $f'(x\_0)=f''(x\_0)= \dots =f^{(n-1)}(x\_0)=0, f^{(n)}(x\_0)\neq 0$, there are now $n$ functional inverses, but most of them represent complex roots around the extremum ($n-1$ if $n$ is odd and $n-2$ if $n$ is even). This hint may be useful if one actually tries to formalize the theorem, since in the complex plane all branches will be included.



"Lagrange inversion" around an extremum
This is easily reduced to the ordinary Lagrange inversion.


Indeed, without loss of generality, $x\_0=0=f(x\_0)=f'(x\_0)<f''(x\_0)$, so that for all real $z$ close enough to $0$ we have $f(z)=z^2 h(z)$, where $h$ is a function analytic near $0$ such that $h(0)>0$. So, the equation $f(z)=w$ (for real $z$ near $0$ and small enough $w\ge0$) can be rewritten as
$$g(z):=z\sqrt{h(z)}=\sqrt w\,\text{sgn}\,z,$$
so that $g$ is analytic near $0$: $g(z)=\sum\_{n=0}^\infty a\_n z^n$ for real $z$ near $0$. The $a\_n$'s can be obtained by Faà di Bruno's formula.


Using now the ordinary Lagrange inversion, we get
$$z=\sum\_{n=0}^\infty b\_n w^{n/2}\text{sgn}^n z$$
for some explicitly written $b\_n$'s and all real $w$ close enough to $0$.


