Suppose I have a power series approximation $y$ to an invertible function $f(x)$, and I know that $y$ convergences around $x$ on an interval $(-R,R)$, $R$ being the radius of convergence.
How are the radii of convergence of the power series $y$ and its reversion (i.e. inverse) related?
Problem:
I have a power series approximation to a function which I cannot invert, so I need the reversion of the power series $y^{-1}$ to approximate the function inverse $f^{-1}$.
If I have the function $f(x)$ on an interval $a \leq x_{0} \leq b$ in the domain $D(f)$, and a power series which converges on that same interval, how can I determine on which interval in the range $R(f)$ the inverse power series will converge?
I obviously want to have a radius of convergence of the inverse power series such that it covers the range $f(a) \leq f(x) \leq f(b)$.
Best Answer
This is an expansion and follow up of my comment.
As mentioned in comment.
$\hspace0.5in$ Without further restriction, there are no relation.
An example is the function $f(x) = x^3$ which is invertible over the real axis and yet its inverse function doesn't have a power series expansion at $x = 0$.
In general, if your function is invertible only over real axis (or part of it), there isn't much one can do but check the power expansions of the function and its inverse function separately.
However, if your function is analytic and injective over some open subset of $\mathbb{C}$ (such a function is known as an univalent function), there is a theorem by Koebe which can help you.
Let's say your function $f$ is univalent over the open disk $B(z_0,R)$. Since $f$ is analytic and locally injective at $z = z_0$, $f'(z_0) \ne 0$. Now define a function $\varphi : B(0,1) \to \mathbb{C}$ by
$$B(0,1) \ni \omega \mapsto \varphi(\omega) = f(z_0 + R \omega ) \in \mathbb{C}$$
We have $\varphi(0) = f(z_0)$ and $\varphi'(0) = R f'(z_0)$. By Koebe quarter theorem, $$f(B(z_0,R)) = \varphi(B(0,1)) \supset B\left( f(z_0), \frac{R|f'(z_0)|}{4} \right)$$
This implies the nearest singularity of the inverse function $\varphi^{-1}$ in $\mathbb{C}$ is at least at a distance $\displaystyle\;\frac{R|f'(z_0)|}{4}$ from $f(z_0)$. As a result, the radius of convergence of the power series of $\varphi^{-1}$ and hence that of $f^{-1}$ at $f(z_0)$ is at least $\displaystyle\;\frac{R|f'(z_0)|}{4}$.