I'm taking my first course in complex analysis and I understand everything but how to find Laurent series. Other than seeing worked examples I can never figure them out myself.
One of the textbook problems I'm stuck on is:
Find the Laurent series expansion of:
$$ \frac{1}{(z^2 – 1)^2} $$
Defined at $|z+1|> 2$
What is the general strategy for approaching this problem? Any hints, guidance or resources would be greatly appreciated.
Best Answer
Express your function in terms fo $z+1$. In this case, you have$$\frac1{(z^2-1)^2}=\frac1{(z-1)^2(z+1)^2}=\frac1{\bigl(2-(z+1)\bigr)^2(z+1)^2}.$$Now, at this point don't deal with the $\dfrac1{(z+1)^2}$ part; leave it to the end. Write $\dfrac1{\bigl(2-(z+1)\bigr)^2}$ as a power series about $-1$:$$\frac1{\bigl(2-(z+1)\bigr)^2}=a_0+a_1(z+1)+a_2(z+1)^2+a_3(z+1)^3+\cdots$$Then the Laurent series that you are interested in is$$\frac{a_0}{(z+1)^2}+\frac{a_1}{z+1}+a_2+a_3(z+1)+\cdots$$