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A contour integral with a third order pole.

[Click here for a PDF version of this post] Here’s problem 31(e) from [1]. Find \begin{equation}\label{eqn:thirdOrderPole:20} I = \int_0^\infty \frac{x^2 dx}{\lr{ a^2 + x^2 }^3 }. \end{equation} Again, we use the contour \( C \) illustrated in fig. 1 fig. 1. Standard above the x-axis, semicircular contour. Along the infinite semicircle, with \( z = R e^{i\theta}…