Let $f(t)$ be a nice Hölder continuous function. Also, suppose that $f$ is even. I'm interested in evaluating integrals of the form:

$$\oint (1-z)^{k+1}\int_0^1 \frac{f(t)}{(1-zt)^{n+1}}dtdz,$$

where for the contour integral, one may assume any loop around $z=1$. Equivalently, I'm interested in calculating Laurent expansions of $F_n(z):=\int_0^1\frac{f(t)}{(1-zt)^{n+1}}dt$ about $z=1$. Note that for $z\in(1,\infty)$, $F_n(z)$ is defined as a Cauchy principle value integral when $n$ is even and as as a Hadamard principle value integral when $n$ is odd. The primary difficulty I'm encountering is that when $z>1,$ the contour integral becomes rather nontrivial. In other words, one needs to understand what's going on as $z$ approaches the real axis to the right of 1. This looks like a sort of Riemann Hilbert problem to me, at least for the evaluation of $F_n(z)$. To this extent, is there a generalization of the Riemann Hilbert method for such double integrals? I mention this because I would like to consider $f_n(t)$, instead of $f(t)$ and then perform asymptotics on the Laurent series coefficeints, as $n\rightarrow\infty$.

Upon swapping integrals, it seems like the problem depends on whether it's a Cauchy or Hadamard integral. Specifically, there is a sharp transition when $1/t$ enters the area bounded by the $\lambda$ contour. This gives me a nonsensical answer that depends on the contour, which is impossible.

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