$$I = \int_0^\infty \frac{x^{-ia}dx}{ x^2 +x+1} = \frac{2\pi}{\sqrt{3}} \frac{\sinh (\pi a/3) }{\sinh (\pi a)} $$
함수 $$f(z) = \frac{z^{-ia}}{z^2 + z + 1} = \frac{e^{-ia \log z}}{ z^2 + z + 1}$$
$z=0$이 branch point이므로 $+x$ 축을 cutline으로 선택하자. 그러면 $0\le \arg(z)\le 2\pi$. 그리고 $z_1=e^{i 2\pi/3}$, $z_2 = e^{i 4\pi/3}$은 simple pole로 residue는 각각
$$ \text{Res}f(z_1) = \frac{e^{-i a \log e^{i 2\pi/3}}} { e^{i2\pi/3}-e^{i 4\pi/3}} = \frac{e^{2\pi a/3}}{i\sqrt{3}}$$
$$ \text{Res}f(z_2) = \frac{e^{-i a \log e^{i 4\pi/3}}} { e^{i4\pi/3}-e^{i 2\pi/3}} = \frac{e^{4\pi a/3}}{-i\sqrt{3}}$$
$C_1$을 따라 $z= x e^{i0}$이므로
$$ \int_{C_1} = \int_0^ \infty \frac{e ^{-ia \log x} dx}{x^2 + x+1} = I$$
$C_2$을 따라 $z= x e^{i 2\pi}$이므로
$$ \int_{C_2} = \int_\infty^0 \frac{e^{-ia (\log x+ {i 2\pi}) } dx}{x^2 + x +1} = - e^{2\pi a} I $$이어서
$$ \int_{C_1 + C_2} = - 2\sinh(\pi a) e^{\pi a} I $$
$C_\epsilon$과 $C_\infty$에서는 0으로 수렴하므로, residue 정리에 의해서
$$ \int_{C_1+C_2} f(z) dz = 2\pi i \times \sum \text{Res}f(z_k) \\ \to~ - 2\sinh(\pi a) e^{\pi a} I = -\frac{4\pi}{\sqrt{3}} e^{\pi a} \sinh \frac{\pi a}{3} \\ \to ~~ I = \frac{2\pi}{\sqrt{3}} \frac{\sinh(\pi a/3) }{\sinh(\pi a)} $$
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