Lemma 4.21 . [045T]

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Lemma 4.21.

These integrals admit the simplified formulae:

\begin{cases}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,\\ I_{02}=-\frac{1}{2}I_{01}+\frac{\pi\sqrt{\mathbb{A}}}{A\varrho|y|_{a}^{2}}\\ I_{03}=\frac{\pi\sqrt{\mathbb{A}}}{A|y|_{a}^{2}}.\end{cases}

Consequently

\begin{cases}I_{p3}=\frac{a_{p\bar{q}}\sqrt{-1}y_{q}}{4\pi A^{1/2}\sqrt{\mathbb{A}}}\frac{1}{\varrho(\varrho+A^{1/2}\mu)},\\ I_{p4}=\frac{a_{p\bar{q}}\sqrt{-1}y_{q}}{4\pi A^{1/2}\sqrt{\mathbb{A}}}\frac{1}{\varrho(\varrho-A^{1/2}\mu)}.\end{cases}