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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 = \omega_0 \pm i a
\end{equation}
one of which sits above the x-axis, one below, and one on the line.

This means that if we compute the usual infinite semicircular contour integral, we have a \( 2 \pi i \) weighted residue above the line and one \( \pi i \) weighted residue for the x-axis pole. That is
\begin{equation}\label{eqn:oscillatorKernel:50}
\begin{aligned}
\oint \inv{ \lr{ z – \omega_0 }^2 + a^2 } \inv{ z – \omega } dz
&=
\lr{ 2 \pi i } \evalbar{ \inv{\lr{z – \lr{ \omega_0 – i a } } \lr{ z – \omega } } }{z = \omega_0 + i a }
+
\lr{ \pi i } \evalbar{ \inv{ \lr{ z – \omega_0 }^2 + a^2 }}{z = \omega } \\
&=
\lr{ 2 \pi i } \inv{\lr{\omega_0 + i a – \lr{ \omega_0 – i a } } \lr{ \omega_0 + i a – \omega } }
+
\lr{ \pi i } \inv{ \lr{ \omega – \omega_0 }^2 + a^2 } \\
&=
\frac{ 2 \pi i }{2 i a} \inv{ \omega_0 + i a – \omega } \frac{ \omega_0 – i a – \omega }{\omega_0 – i a – \omega }
+
\lr{ \pi i } \inv{ \lr{ \omega – \omega_0 }^2 + a^2 } \\
&=
\frac{ \pi }{ \lr{ \omega – \omega_0 }^2 + a^2 } \lr{ \frac{\omega_0 – \omega}{a} – i + i },
\end{aligned}
\end{equation}
or
\begin{equation}\label{eqn:oscillatorKernel:100}
\boxed{
I =
\frac{ \pi \lr{ \omega_0 – \omega } }{ a \lr{ \lr{ \omega – \omega_0 }^2 + a^2} }.
}
\end{equation}

Interestingly, Mathematica doesn’t seem to be able to solve this integral, even setting PrincipleValue to True. The solution ends up with a bogus seeming \( \textrm{Im}\left(\omega_0-\omega \right) = \textrm{Re}(a) \) restriction, and as far as I can tell, the Mathematica result is also zero after simplification that it fails to do. Mathematica can solve this if we explicitly state the PV condition as a limit, as shown in fig. 1.

fig. 1. Coercing Mathematica to evaluate this.

References

[1] F.W. Byron and R.W. Fuller. Mathematics of Classical and Quantum Physics. Dover Publications, 1992.