So far in this series, I have used three different techniques to show that

.

For the third technique, a key step in the calculation was showing that the residue of the function

at the point

was equal to

.

Initially, I did this by explicitly computing the Laurent series expansion about and identifying the coefficient for the term .

In this post and the next post, I’d like to discuss alternate ways that this residue could have been obtained.

Notice that the function has the form , where and are differentiable functions so that and . Therefore, we may rewrite this function using the Taylor series expansion of about :

Therefore, the residue at is equal to , or the constant term in the Taylor expansion of about . Therefore,

For the function at hand and . Therefore, the residue at is equal to , matching the result found earlier.

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