So the idea is that you can represent a function as a product of its zeroes, and there are some fundamental factors that often crop up.
I am interested in, give this is the WF of sine :
Is it really this simple? Just take a function where it is zero, and product these? For $sin(\pi)$ with $z^2$ = $1$ the first factor in the product is 0. And for any integer multiple of $\pi$, $n$, the $n$th term in the product will be zero, and the correct result is achieved. But how does knowledge of the zeroes of the function determine the unique sinusoidal curve between those zeroes? Is it due in this case to $\pi$ giving the sinusoidal shape and in general, what is the determination of the shape of the curve between the zeroes that allows the WF to reproduce it?
Other questions is there a neat formula that relates WF to Fourier Series, or Taylor Expansion? (other methods of approximating functions).
And the bonus question is : can we form the Riemann Zeta function by the WF of a product of its zeroes?