I need to compute a series expansion in $\epsilon$ (at least to second order) of integrals like
$$g(\epsilon):=\int_0^\epsilon f(\epsilon,v)\, dv,$$ as $\epsilon \to 0^+$, where the functions in question are very complicated. (Too complicated, in my opinion, to express as Taylor series themselves.)
Does anyone have a reference for such things? I have tried naively expanding around $0$, but some of the derivatives blow up there, so I haven't been able to reconcile terms like $g''(0)$ which don't exist.
(The derivatives at $\epsilon=0$ are functions of $v$. But, they are not integrable on $[0,\epsilon]$)
Thanks.
EDIT: per request, here is one such $f$:
Note that we have $|v|<\epsilon$ and $\epsilon>0$ small.
EDIT #2: Would still like a reference for the theory of integral expansions if someone has one they like.