construct entire function

Question

Can anyone please explain to me how to use Weierstrass product theorem to construct entire function? If I want to construct entire function with simple zeros on the positive real axis at the points $\sqrt n , n\ge1$ and double zeros on the imaginary axis at the points $ \pm i \sqrt n , n\ge1$ with no other zeros.

Answer 1

Enumerate your zeros, counting multiplicities, as a single sequence $a_n$, $n=1,2,3,\ldots$. In your case, you might do this as $$\sqrt{1},\; i\sqrt{1},\; i\sqrt{1},\; -i\sqrt{1},\; -i\sqrt{1},\; \sqrt{2}, i\sqrt{2},\; \ldots$$ Now Weierstrass says you take nonnegative integers $p_n$ such that $$ \sum_n (r/|a_n|)^{1+p_n} < \infty$$ for all $r > 0$. In this case we note that $|a_n|$ goes to $\infty$ essentially as $\sqrt{n}$. Thus $p_n = 2$ will do, because $\sum_n r^3/n^{3/2}$ converges. So our Weierstrass product could be $$ \prod_{n=1}^\infty E_2(z/a_n) = \prod_{n=1}^\infty \left(1 - \frac{z}{a_n}\right) \exp\left(\frac{z}{a_n} + \frac{z^2}{2a_n^2}\right) $$

Answer 2

let be the sets of zeros $ x_i $ of a function $ f(x_i) $ then the entire function can be constructed in the form

$$ f(x)= \prod _{n=0}^{\infty}(1-\frac{x}{x_{i}}) $$

then this $ f(x) $ is an entire function, see the product

$$ \prod _{n=0}^{\infty}x_{i}$$

must be zeta regularizable

Answer 3

There is a trick to the Weierstrass product. If $a_k$, $k\geq 1$ is a sequence tending to zero then $$\prod_{k\geq 1} (1-a_k z)$$ need not be convergent. You may, however, recover convergence by multiplying by a factor that has no zeros. You start by looking at $$1/(1-a_k z) = \exp( -\ln (1-a_k z))=\exp(\sum_{j\geq 1} \frac{(a_kz)^j}{j})$$ but you truncate the sum in the exponential to a finite order $p_k$. By choose suitable $p_k$ (which may tend to infinity) the product:$$\prod_{k\geq 1} (1-a_k z) \exp\left(\sum_{j= 1}^{p_k} \frac{(a_kz)^j}{j}\right) $$ is an entire function.