I assume an integral $$I=\int_0^\infty f(x)g(x)\mathrm dx \tag{1}$$
where the function $f(x)$ can be represented as a contour integral in complex plane:
$$f(x)=\oint_\Delta \frac{h(x,s)}{s^n(1-s)}\mathrm ds\tag{2}$$
with $h(x,s)$ having no poles other then $s=0$ and the contour $\Delta$ is any contour (usually a circle with raduis $0<r<1$) which encircles the origin $s=0$.
So changing the order of integration I get $$I=\int_0^\infty \left(\oint_\Delta \frac{h(x,s)}{s^n(1-s)}\mathrm ds \right) g(x)\mathrm dx=\oint_\Delta \frac{1}{s^n(1-s)}\left(\int_0^\infty h(x,s) g(x)\mathrm dx\right)\mathrm ds
\tag{1*}$$
Let's assume that the inner integral can be evaluated too: $I_{\text{inner}}(s)=\int_0^\infty h(x,s) g(x)\mathrm dx$ but in addition to the poles at $s=0$ and $s=1$ it has $2$ new poles at $s=\alpha_1$ and $s=\alpha_2$ which are inside the unit circle and hence can be possibly inside (both or each of them individually) or outside the contour of integration $\Delta$.
Let's in addition assume that the resultant integral $I=\oint_\Delta \frac{1}{s^n(1-s)}I_{\text{inner}}(s)\mathrm ds$ can be evaluated (and gives four different answers $I_{\alpha_1}, I_{\alpha_2}, I_{\text{none}}, I_{\alpha_1\,\alpha_2}$) for all the cases: $\alpha_1$ or $\alpha_2$ is inside $\Delta$; neither of them are inside $\Delta$ and both of them are there.
But which of them should I choose since $\Delta$ is an arbitrary contour?
So I'm tackled by the following questions:
- How should I cope with new poles arising after evaluating $I_{\text{inner}}(s)$?
- Should I choose $\Delta$ beforehand somehow keeping in mind the possibility of new poles?
- What bothers me most is the step with interchanging the order of integration. What are the necessary and sufficient conditions for changing the order of a real and a contour integral?