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- #1

\(\DeclareMathOperator{\Res}{Res}\)

Given

\[

\Ima\left[\int_{-\infty}^{\infty}\frac{e^{iz}}{z(\pi^2 - z^2)}dz\right].

\]

I know the integral is equal to

\[

2\pi i\sum_{\text{UHP}}\Res(f(z); z_j) + \pi i\sum_{\mathbb{R}\text{ axis}}\Res(f(z); z_k).

\]

However, the poles are \(z = 0\) and \(z = \pm\pi\) which are all on the real axis so we just have the sum on the real axis.

\[

\pi i\sum\lim_{z\to z_j}(z - z_j)\frac{e^{iz}}{z(\pi^2 - z^2)} =

\pi i\left[\frac{1}{\pi^2} + \frac{1}{2\pi^2} - \frac{1}{2\pi^2}\right] = \frac{i}{\pi}

\]

However, the solution is \(\frac{2}{\pi}\). What is wrong?