Can I apply Lagrange inversion theorem? [closed]

Question

I want to invert the equation $$\eta = g(x)\sqrt{1+g'(x)^2}$$ to get $x$ as a function of $\eta$. Assume $g(0)=0$, $g'(0)=0$ and $g'(x)>0$ for $x>0$ (Think $g(x) = x^p$ for $p\geq 2$ integer). Define $$ f(x) = g(x)\sqrt{1+g'(x)^2}, $$ then I need to invert $f$ to get $x$. My question is: if I have a Taylor expansion around $0$ of $f$, can I get a Taylor expansion of $x$ in $\eta$ around 0? Can I use Lagrange inversion formula to do that? I looked it up online but one of the assumption if $f'(0)\neq0$ which I don't have. I'm not familiar with the Lagrange inversion theorem so I don't know how that assumption is important or how to get around it.

Answer 1

If $g$ is analytic at $0$ with a zero there of order $p$, then $f$ is analytic at $0$ and also has a zero there of order $p$. The series of $x$ in terms of $\eta$ is then a Puiseux series, $\sum_{j=1}^\infty c_j \eta^{j/p}$. To obtain it, use Lagrange inversion on $\eta^{1/p} = f(x)^{1/p}$, which has an ordinary Taylor series.