If we have an analytic function but which has a branch point (yes technically it's only analytic in open sets which are disjoint with a branch cut but that's besides the point) is there a way to expand the function around the branch point? In particular there's a function that comes up in thermal field theory, which is the one-loop thermal contribution to the effective potential (if anyone knows any QFT)
$f(y) = \int_0^\infty x^2\log(1-e^{-\sqrt{x^2+y^2}})=-\frac{\pi^2}{90}+\frac{y^2}{24}-\frac{y^3}{12\pi}-cy^4(\log(y)+d)+O(y^6)$
where c,d are some constants. A derivation of the above expansion for small $y$ can be found in a paper by Dolan and Jackiw called Symmetry Behaviour at Finite temperature.
They use techniques from dimensional regularisation but I was wonder if there was any formalism in complex analysis which allowed one to expand sufficiently nice complex functions around a branch point in terms of elementary ones.
This is not a full answer but I've discovered that there is a type of series called a Puiseux series which allows functions with branch points in the series. Most sources state that a Puiseux series a power series where each term in the series looks like $x^i$ where $i$ can be a rational number whose denominator is bounded by some integer $n$; this basically means $i=k/n$ for some fixed $n$ and any integer $k$. However, WolframMathworld also allowed logarithms in the expansion which may be multiply nested, e.g. $\ln\ln x$. This seems to be contrary to most other sources however.
I've not discovered so far how to calculate the Puiseux series. If one precludes logarithms from being in the series, if $f(x)$ admits a Puiseux expansion at zero then $f(x^n)$ will be analytic (since $n$ bounds the denominators of the exponents). However they seem to have more relevance in polynomial ring theory and algebraic geometry than complex analysis.