# Thread: infinite product, entire function

1. ## infinite product, entire function

Construct an entire function that has simple zeros on the real axis at the points $\displaystyle \pm n^{\frac{1}{4}}, n \geq 0$, and no other zeros.

The back of the book says that $\displaystyle z \prod_{n=1}^{\infty} (1-\frac{z^2}{\sqrt{n}})e^{\frac{z^2}{\sqrt{n}}+\frac {z^4}{2n}}$ works. However, I do not see how to show that this function is entire and satisfies the other conditions. In this section, we covered the Weierstrass Product Theorem; however, I do not think we need to use that here. I need help with this one. Thanks.

2. Originally Posted by Erdos32212
Construct an entire function that has simple zeros on the real axis at the points $\displaystyle \pm n^{\frac{1}{4}}, n \geq 0$, and no other zeros.

The back of the book says that $\displaystyle z \prod_{n=1}^{\infty} (1-\frac{z^2}{\sqrt{n}}e^{\frac{z^2}{\sqrt{n}}+\frac{ z^4}{2n}})$ works. However, I do not see how to show that this function is entire and satisfies the other conditions. In this section, we covered the Weierstrass Product Theorem; however, I do not think we need to use that here. I need help with this one. Thanks.
You seem to have the parentheses in the wrong place. Surely the answer should be $\displaystyle z \prod_{n=1}^{\infty} \Bigl(1-\frac{z^2}{\sqrt{n}}\Bigr)e^{\frac{z^2}{\sqrt{n}}+ \frac{z^4}{2n}}$. That function clearly has zeros at the points $\displaystyle \pm n^{1/4}$, provided that the infinite product converges. The convergence of the infinite product is presumably where the Weierstrass theorem comes in.