Find the zeros of $z^2\sin z$ and establish their order

Question

Find the zeros of $z^2\sin z$ and establish their order

The zeros of $z^2 \sin z$ are clearly given by $z_1 = 0$ and $z_n = n \pi$. Given this how could I go about finding the order of these zeros?

My notes do not not give examples of how one could find the order of theze zeros and Wikipedia hasn't helped me much either.

Answer 1

Let $f(z)=z^2 \sin z$.

Then $f(0)=f'(0)=f''(0)=0$ and $f'''(0) \ne 0$. Hence $f$ has at $0$ a zero of order $3$.

For $n \ne 0$ we have $f(n \pi)=0$ and $f'( n \pi) \ne 0$. Hence $f$ has at $n \pi $ a zero of order $1$.

Answer 2

Hint: The order of the zero of $f(z)g(z)$ at $z=z_0$ is equal to the order of the zero of $f(z)$ plus the order of the zero of $g(z)$ (where poles have negative order, and a finite, non-zero value is a zero of degree $0$).