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Thread: infinite product, entire function

  1. #1
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    infinite product, entire function

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

    The back of the book says that 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.
    Last edited by Erdos32212; Apr 18th 2010 at 11:38 AM.
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  2. #2
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    Quote Originally Posted by Erdos32212 View Post
    Construct an entire function that has simple zeros on the real axis at the points \pm n^{\frac{1}{4}}, n \geq 0, and no other zeros.

    The back of the book says that 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 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 \pm n^{1/4}, provided that the infinite product converges. The convergence of the infinite product is presumably where the Weierstrass theorem comes in.
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