So I learned in my complex analysis class the concept of Laurent series but my teacher's approach was fully theoretical: no actual example and no example on how to find a Laurent series for a function on a specific annulus.
So when I started doing some exercises I obviously found it very difficult and frustration because I have no idea how to proper tackle the problem.
My first exercise is the following:
Find the Laurent series of the functions: $$f(z)=\frac{\sin(z)}{z^2}\ \ \ \ \text{and}\ \ \ \ g(z)=\frac{1}{3-z}$$ on the annulus $0 < |z| < 3$ and $|z| > 3$
Knowing the theory behind Laurent series, my first thought, was to apply the formula:
$$c_n=\frac{1}{2\pi i}\oint_\gamma \frac{f(z)}{(z-a)^{n+1}}dz$$
I think this may be possible for function $g$ but I don't see it working that easily for function $f$.
So my question is: What is the most appropriate way to find the Laurent series of this functions? What are the basic things you look for in a function in other to make the process of finding a Laurent series more easy. How does the different annulus chance you approach for solving the problem?
Basically I am asking you to teach me some tips and teach me the mindset behind finding Laurent series and the intuition that my teacher should have have in his lecture.