複素積分4

今回はこの積分の解説をします
https://twitter.com/integralsbot/status/1433409545770962949?s=21

$I=\displaystyle\int_0^{2\pi}\frac{x}{\phi-\cos^2x}dx$
KingPropertyを使って
$2I=\displaystyle\int_0^{2\pi}\frac{2\pi}{\phi-\cos^2x}dx$
$I=\displaystyle\pi\int_0^{4\pi}\frac{1}{\sqrt{5}-\cos{t}}dt$
$=\displaystyle 2\pi i\oint_{C:z=e^{it}}\frac{2}{z-2\sqrt{5}z+1}dz$
$=\displaystyle -4\pi^2・\left.\frac{2}{z-\sqrt{5}-2}\right|_{z=\sqrt{5}-2}$
$=\displaystyle 2\pi^2$