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I'm wondering how to solve the integral $\int_\mathbb{R}\frac{{}_2F_1\left(\Delta, 2\Delta + \frac{d - 1}{2}; 2\Delta, 1 - \frac{(x - z)^2}{(z - y)^2}\right)}{|z|^{2\Delta}|z - y|^{4\Delta - d + 1}}dz$?

I think the first step to do is to rewrite $|z| = \left(z^2\right)^{1/2}$ and $|z - y| = \left[(z - y)^2\right]^{1 / 2}$ so I don't need to split the integral into several intervals. After this I think that with an appropriate variable change this could be recognized with the integral representation of a ${}_3F_2$-Hypergeometric integral, i.e. ${}_3F_2\left(a_1, a_2, c_1; b_1, c_2; x\right) = \frac{\Gamma(c_2)}{\Gamma(c_1)\Gamma(c_2 - c_1)}\int_0^1t^{c_1 - 1}(1 - t)^{c_2 - c_1 - 1}dt{}_2F_1\left(a_1, a_2; b_1; tx\right)$.

I would be very glad if you gave this integral a try.

For the interested reader this integral appeared for me when calculating a two-point correlator close to a boundary in a conformal quantum field theory.

Edit:

$d$ is the spacetime dimensions, i.e. I consider $d - 1$ spatial dimensions and one time dimension. I consider d to be between three and four, e.g. $d = 4 - \epsilon$ or $d = 3 + \epsilon$, where $\epsilon$ is a parameter that may regularize the integral.

$\Delta$ is called the scaling dimension. I'm most interested in the case when $\Delta = \frac{d - 2}{2} = 1 - \frac{\epsilon}{2}$.

It would be optimal to solve the integral for general $d$ and $\De$, but that may be too much to ask.

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  • $\begingroup$ In order to avoid addressing unnecessary casework, it would be helpful if you could specify the conditions on the four parameters $\Delta,d,x,y$. Since the context of the problem is cQFT, I'm gonna guess that $d$ stands for a spatial dimension and so is probably a positive integer. Also, for the sake of the convergence of the $_3F_2$ integral you provided, it would be very nice if we could assume $0<\Delta<\frac12$. $\endgroup$ – David H Oct 12 '18 at 0:20
  • $\begingroup$ Oh, that's a good idea. I edit in this information in my original post as well. $d$ is the spacetime dimensions, i.e. I consider $d - 1$ spatial dimensions and one time dimension. I consider d to be between three and four, e.g. $d = 4 - \epsilon$ or $d = 3 + \epsilon$, where $\epsilon$ is a parameter that may regularize the integral. $\De$ is called the scaling dimension. I'm most interested in the case when $\De = \frac{d - 2}{2} = 1 - \frac{\epsilon}{2}$. It would be optimal to solve the integral for general $d$ and $\De$, but that may be too much to ask. $\endgroup$ – A.Dunder Oct 15 '18 at 7:04

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