Try expanding as a series around , namely you should have an easy time then, no? By the way I hope you know..bam!
I want to calculate the residue of at .
is a pole of order 2 since
So the standard approach is
After differentiating and applying L'Hospital's rule once, the limit is still indeterminate and very messy. Is there a better approach than repeated applications of L'Hospitals rule?
Here's what I did. It probably could have been done in less steps.
I wrote the function as . Then I found the Taylor series of centered at , and used synthetic division to find the Laurent series for centered at . Next I found the Taylor series for and centered at , and again used synthetic division to find the Laurent series for centered at . Finally I multiplied the Laurent series of by the Laurent series for .
With the substitution the problem is finding the residue at of...
(1)
First we indagate on that can be written as...
(2)
... and from (2) You can derive that is...
(3)
Now we indagate on that can be written as...
(4)
... and from (4) You can derive that is...
(5)
... so that is...
(6)
Kind regards