This series was inspired by a question that my wife asked me: calculate
Earlier, I evaluated this last integral using partial fractions, separating into the cases , , and . Now, I’ll calculate this same integral using contour integration. (See Wikipedia and Mathworld for more details.)
It turns out that can be rewritten as
where is the contour in the complex plane shown above (graphic courtesy of Mathworld). That’s because
To show that the limit of the last integral is equal to 0, I use the parameterization , so that :
The above limit is equal to zero because the numerator grows like while the denominator grows like . (This can be more laboriously established using L’Hopital’s rule).
Therefore, I have shown that
and this contour integral can be computed using residues.