In the context of a Borel transform relevant to a problem in QFT (http://arxiv.org/abs/0806.3156 -- Appendix A, i.p. Eq. A6 and Eq. A7) I've encountered a particularly confusing integral and am struggling to reproduce the results of the authors for my own work.
The integral in question has the following form: $$ \int_0^\infty dt \frac{\exp(-t/\alpha)}{(p - bt)^q}, $$ where $\alpha, b$ are positive real values, p is some positive integer, and $q$ some real value.
With the branch cut beginning at $t = \frac{p}{b}$, a principal value prescription is adopted to assign a value to the above, by indenting the contour above (or below) the beginning of the branch cut and ultimately taking the relevant limit.
It seems that I can without issue reproduce the term proportional to the generalisation of the exponential integral $E_q$ that appears in Eq. A7 of the above reference as a consequence of integrating along the real axis from $0 \to \frac{p}{b} - \epsilon$, then from $\frac{p}{b} + (1 + i)\epsilon \to \infty$, then taking the limit $\epsilon \to 0$.
The term proportional to $\Gamma(1 - q)$ seems to arise from the indented portion of the contour, however, and eludes me. I tried naïvely parametrising this part of the contour as $t = \frac{p}{b} + \epsilon e^{i \theta}$ for $\theta$ ranging from $\pi$ to $0$ (for indenting above the cut; from $\pi$ to $2\pi$ for indenting below), but only succeeded in reaching expressions that seemed to vanish in the limit $\epsilon \to 0$.
A standard semicircular contour breaks down due to the divergence of the integrand as $t$ becomes abitrarily large and negative, and I was unable to find any success in using the Hankel contour for this problem.
Would anybody be able to point me in the direction the authors of the above might have taken to obtain their result?