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In the following Fox H-function the contour $L$ is either $L_{-\infty}$, $L_{+\infty}$ or $L_{i\gamma\infty}$.

$$ H_{p,q}^{\,m,n} \!\left[ z \left| \begin{matrix} ( a_1 , A_1 ) & ( a_2 , A_2 ) & \ldots & ( a_p , A_p ) \\ ( b_1 , B_1 ) & ( b_2 , B_2 ) & \ldots & ( b_q , B_q ) \end{matrix} \right. \right] = \frac{1}{2\pi i}\int_L \frac {(\prod_{j=1}^m\Gamma(b_j+B_js))(\prod_{j=1}^n\Gamma(1-a_j-A_js))} {(\prod_{j=m+1}^q\Gamma(1-b_j-B_js))(\prod_{j=n+1}^p\Gamma(a_j+A_js))} z^{-s} \, ds $$

Suppose $L=L_{i\gamma\infty}$. According to The H-Function: Theory and Applications by A.M. Mathai, $L=L_{i\gamma\infty}$ is a contour starting at the point $\gamma-i\infty$ and going to $\gamma+i\infty$ where $\gamma\in R=\left(-\infty, +\infty\right)$ such that all the poles of $\Gamma\left(b_j+B_js\right), j=1,\ldots,m$ are separated from those of $\Gamma\left(1-a_{\lambda}-A_{\lambda}s\right), \lambda=1,\ldots,n$.

My question is how to determine $\gamma$?

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Suppose the integrand is $\Gamma(-1/2 - s) \Gamma(s) z^{-s}$. The requirements are that the contour $\mathcal L$ separate the poles and that the integral over $\mathcal L$ converge.

As long as the requirements hold, you can deform the contour freely. In this example, you can deform the vertical parts of $\mathcal L$ to horizontal rays going to $-\infty \pm i \gamma$ if $|z| \leq 1$ and to horizontal rays going to $+\infty \pm i \gamma$ if $|z| \geq 1$. This can be useful for extending a function defined on the unit disk.

There is a case where the integrals over the left and the right loops both exist but are different (the contours cannot be deformed into one another). But since this is the case where the function is discontinuous on $|z| = 1$, apparently no one is terribly interested in how it is defined at the discontinuity.

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