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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 : enter image description here

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?

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2 Answers 2

No, just the zeroes are not enough to determine the "entire" function. For instance, we can just take $e^{z/n + z^2/2n^2}$ instead of $e^{z/n}$ (in the product there) and get a different function with the same set of zeroes.

But, by the Hadamard factorization theorem, given the (finite) order of the function, the infinite product portion (called the canonical product) of the factorization can indeed be made to match: i.e. if two functions have the same set of zeroes(counting multiplicity) and the same finite order, then their ratio is $e^{P(z)-Q(z)}$, where $P$ and $Q$ are finite polynomials of the same degree.

I guess you can intuitively think of them as infinite polynomials, which are indeed determined by their roots, upto a "constant" term. In order to make the infinite product converge, Weierstrass came up with the idea of introducing the $E_{n}$.

All this does not apply to Riemann-Zeta function as it has a pole $z=1$, and the factorization theorems assume the function is entire.

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I admit, I am a bit rusty though... –  Aryabhata Apr 19 '13 at 2:07

Look at the paper written by Richard Stanley, EJC 18(2) (2011), #p16.

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Can you please give more details? What is the paper about? How does it help here? –  Aryabhata Apr 19 '13 at 1:44

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