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I am reading the article arXiv: 2207.09961, there are some interesting elliptic integrals, i.e. the formula (3.7) and (3.8). You can also see this image

enter image description here

where $p_0(z)=\sqrt{-Q_0(z)}$ and $Q_0(z)=-\frac{\omega^2\ L^2(z^4+1)-z^2(l+2)^2}{z^4}$. It is a period integral as the author said. However I have no good idea to derive it. Could you tell me some hints? Thank you very much!

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    $\begingroup$ Better to type out the equation in full, than have an image like this $\endgroup$
    – David Roberts ♦
    Mar 5, 2023 at 21:01
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    $\begingroup$ Check DLMF Ch.19 a handbook reference on Elliptic Integrals. $\endgroup$
    – Igor Khavkine
    Mar 6, 2023 at 0:14
  • $\begingroup$ Thank you for your comment. @David Roberts $\endgroup$
    – amon Hsu
    Mar 6, 2023 at 1:07

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