Finding residue for $z_0 = 0$ in $f(z) = {{\sin z - z}\over {z^3( z^2 + 9z)}}$

Question

I need (just) a hint for finding residue by either using Laurent series or by using limit formula for poles of kth order. I do realize that $z_0 = 0$ is a simple pole, but limit formula is not quite elegant when using it.

Answer 1

One might recall that

$$\sin z=z-\frac16z^3+\frac1{120}z^5-\dots$$

Thus, you should be able to see that

$$\frac{\sin z-z}{z^4(z+9)}=\frac1{z+9}\left(-\frac16\frac1z+\frac1{120}z-\dots\right)$$

which is pole of order one at $z=0$.