I am taking a class on using perturbation theory methods to get approximate solutions to certain ODEs in which there is a small parameter $\epsilon$. Often one starts out by assuming the solution is an analytic function of $\epsilon$ and this allows you to write the solution as a power series in $\epsilon$. However sometimes this (slightly mysteriously) "doesn't work" and instead a non-integer power expansion does the job.
My question is this. For smooth functions we have Taylor's theorem which says we can write that function as an (integer) power series. What is the corresponding theorem for non-integer power series? What is the class of functions that can be expanded in this way?
You can have Puiseux series which are essentially just $\;f(x) := g((x-x_0)^{1/n})\;$ for some function $g$ which has a Laurent series expansion. The key fact is that $\;f(x)\;$ locally at the point $x_0$ does not behave like a polynomial. It behaves like $x^\alpha$ where $\alpha$ is negative or not an integer. The first step is to determine the value of $\alpha$ if it exists.