Bounding an Error on the inverse.

Question

Suppose $f(x) = x^p + $ lower order terms. Then Asymptotically $f^{-1}(x) \sim x^{\frac{1}{p}} $ for large $|x|$. How can we bound the error in this asymptotic approximation in terms of $|x|$

Answer 1

$f^{-1}(f(x))$ is supposed to be $x$. You can look at different kinds of errors, the two simplest are the absolute difference and the relative difference: $$E_1(x) = x- f^{-1}(f(x)),\qquad E_2(x) = 1- \frac{f^{-1}(f(x))}{x}$$ If you want to find an asymptotic expansion of the error at infinity you need to look at $E(1/x)$ for $x$ close to $0$. Then you can do a Taylor expansion. This is nicer if we use $E_2$: $$E_2(1/x)=1-x\sqrt[p]{\frac1{x^p}+\sum_{k<p} a_k\frac1{x^k}}=1-\sqrt[p]{1+\sum_{k<p}a_kx^{p-k}}$$ Here the $a_k$ are the coefficients in $f(x) = x^p + \sum_{k<p} a_k x^k$. This term is $0$ at $x=0$, the first derivative is some function you can easily derive. At $0$ the derivative evaluates to: $$-\frac1p a_{p-1}$$ so you have $$E_2(1/x) = 0 -\frac{a_{p-1}}{px} + O(\frac1{x^2})$$ continue in this way.