Multiplicity of zeros Can you explain me how to get the multiplicity of a zero? 
In particular, I would ask you how to determine the zeros' multiplicity of
$$\cos(\frac{\pi}{2}z)$$
I suppose they are $z = 2k+1, k \in \mathbb{Z}$ and are simple zeros for the function.

It suffices to show that the zero at $z=1$ is simple, i.e. that
$$\frac{\cos(\pi/2 z)}{z-1}$$ is a non-zero constant when $z\rightarrow 1$.
To do that, we use L'Hospital's rule to compute the limit to be $-\pi/2 \sin(\pi/2)=-\pi/2\not=0$.
For the other zeros this works the same way.
In general, an analytic function $f$ has a zero of multiplicity $k$ at $z_0$ if there exists an analytic function $g$ defined in some neighbourhood of $z_0$ with $g(z_0)\not=0$ such that
$$f(z)=(z-z_0)^k g(z)$$
close to $z_0$. In our example, the above quotient is that function $g$ and the calculation we did was to check that $g(z_0)\not=0$.
