To evaluate these integrals, we introduce a radial variable

$$s=a_{p\bar{q}}(s_{p}+\sqrt{-1}y_{p})(s-\sqrt{-1}y_{q}),$$

and then elementary calculations in polar coordinates give

$$\begin{split}I_{01}=\frac{\pi}{\sqrt{\mathbb{A}}}\int_{\frac{A}{\mathbb{A}}|y|_{a}^{2}}^{\infty}\frac{1}{s(s+A\mu^{2})^{3/2}}ds,\end{split}$$

and similarly

$$\begin{split}I_{02}=\frac{\pi}{\sqrt{\mathbb{A}}}\int_{\frac{A}{\mathbb{A}}|y|_{a}^{2}}^{\infty}\frac{1}{s^{2}(s+A\mu^{2})^{1/2}}ds=-\frac{1}{2}I_{01}+\frac{\pi\sqrt{\mathbb{A}}}{A\varrho|y|_{a}^{2}},\end{split}$$

together with the formula for $I_{03}$. The formulae for $I_{p3}$ and $I_{p4}$ follow from taking linear combinations. ∎