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, I’d like to discuss another way 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 :

Clearly,

Therefore, the residue at can be found by evaluating the limit . Notice that

,

where is the original denominator of . By L’Hopital’s rule,

.

For the function at hand, and , so that . Therefore, the residue at is equal to

,

matching the result found earlier.

### Like this:

Like Loading...

*Related*

## 1 Comment