Laurent series of $e^z/(z^3+z)$

Question

Find the first four terms of the Laurent series of $$\frac{e^z}{z^3+z}$$ in the punctured disk $0<|z|<1$.

We can apply the formula to get $$f(z)=\sum_{k=0}^\infty a_kz^k$$ where $$a_k=\dfrac{1}{2\pi i}\int_{|z|=1}\dfrac{f(z)}{z^{k+1}}dz$$ But it seems hard to evaluate those values (say, by residue theorem.) How can we proceed?

Answer 1

Write the function in the form

$$f(z) = \frac1z\cdot e^z \cdot \frac{1}{1+z^2}.$$

Use the Taylor expansions of the last two factors. Multiply until you know the first four terms.