calculating the residue of $f(z)=\frac{\cot(z)\coth(z)}{z^3}$

Question

Find the residue of

$$f(z)=\frac{\cot(z)\coth(z)}{z^3}$$ at $z=0$.

I tried to use the Cauchy's residue formula, but the calculation is very complex. Can it be done in a simpler way?

Answer 1

We use the Laurent series of $\cot$ and $\coth$ around $0$. See this link .

If we take $$\coth (z)=\frac 1z+\frac z 3-\frac{z^3}{45}+o(z^4)$$ $$\cot(z)=\frac 1z-\frac z 3-\frac{z^3}{45}+o(z^4)$$

Now use these expressions to calculate $\frac{\cot z \coth z}{z^3}$, the coefficient of $z^{-1}$ is by definition the residue of that expression.