# Principal value of hypergeometric function

1. Dec 7, 2015

### CAF123

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!

2. Dec 12, 2015

### Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?