I have encountered the following question:
Question: Consider the Laurent expansion
$$ \dfrac{e^{z}}{\cos(2z)} = \sum_{n = - \infty}^{\infty} a_{n}z^{n}, \qquad \dfrac{\pi}{4} < \vert z \vert < \dfrac{3\pi}{4}. $$
Determine the coefficients $a_{-1}$ and $a_{1}$.
What I know: It's enough to find the Laurent expansion of $\dfrac{1}{\cos(2z)}$ because the expansion of $e^{z}$ is clear, but in all examples that I have done before, the range for $ z $ is something like $ \vert z - \alpha \vert < r $. But here, this range is different. Can anyone help me solve that?
Thanks in advance.
Hint : $a_{-1}$ is the Residue of $f(z) = \frac{e^{z}}{cos(z)}$ but if $f(z) = \frac{p(z)}{q(z)}$ with $p,q$ holomorphic function and $z_{0}$ is a simple zero of $q$ such that $p(z_{0}) \ne 0$ Res$(f,z_{0}) = \frac{p(z_{0})}{q'(z_{0})}$