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A PV integral using contour integration.

[Click here for a PDF version of this post] Here’s the second last real-integral sub-problem from [1], problem 31(j). Find \begin{equation}\label{eqn:oscillatorKernel:20} I = P \int_{-\infty}^\infty \inv{ \lr{ \omega’ – \omega_0 }^2 + a^2 } \inv{ \omega’ – \omega } d\omega’. \end{equation} Our poles are sitting at \( \omega \), and \begin{equation}\label{eqn:oscillatorKernel:80} \alpha, \beta =…