In my textbook, as part of an example, I am given that " is its own Laurent expansion about z = 1, where it has a double pole."
My reasoning is this:
If we define a punctured disc centred at 1 with radius r > 0, then we can express it as a Laurent series , where . Having a double pole means that , while , for all i < -2.
How am I supposed to compute the coefficients of the Laurent series? Am I supposed to use the formula for described above? All I seem to be able to do is the following:
, which does not equal zero (what I am trying to show), and, for example, , which does equal zero (what I am trying to show).
Can anyone shed some light on this?