I am looking to write the hypergeometric function $${}_2F_1\left(1,1,2+\epsilon, -\frac{\alpha}{\beta}\right) = \int_0^1\,dt\,\frac{(1-t)^{\epsilon}}{1-tz + i\delta},$$ where ##z=-\alpha/\beta## and ##0< \beta < - \alpha##, in terms of its real and imaginary part. The ##i\delta## prescription is to shift the denominator away from the pole at ##t=1/z##. I know that $$\frac{1}{1-tz+i\delta} = \text{P.V} \frac{1}{1-tz} -i\pi \delta(1-tz)$$ so to compute the real part I am left with the problem with finding $$\text{P.V}\int_0^1\,dt\,\frac{(1-t)^{\epsilon}}{1-tz}$$ I tried writing this as $$\lim_{\tau \rightarrow 0} \left(\int_0^{1/z-\tau} + \int_{1/z+\tau}^1\right)\frac{(1-t)^{\epsilon}}{1-tz} dt$$ but I am not sure how to progress. I tried using the residue theorem and coming up with a closed contour but the limits do not extend to ##\pm \infty##. Any help would be great, thanks!(adsbygoogle = window.adsbygoogle || []).push({});

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# Principal value of hypergeometric function

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