Zeroes of analytic functions

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  • #1
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Suppose I have an analytic function, [itex]f[/itex], with radius of convergence [itex]1[/itex]. From http://planetmath.org/encyclopedia/ZerosOfAnalyticFunctionsAreIsolated.html" [Broken] it follows that any [itex]0[/itex] of [itex]f[/itex] in [itex] \{ z : |z| < 1 \}[/itex] is isolated. But is it possible that the zeroes have a limit point on the boundary of convergence? For example, is possible that an analytic function with radius of convergence 1 has zeros exactly at [itex]1-\frac{1}{n}[/itex] for all [itex]n \in Z^{+}[/itex]?
[Edit: Added question marks.]
 
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  • #2
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Suppose I have an analytic function, [itex]f[/itex], with radius of convergence [itex]1[/itex]. From http://planetmath.org/encyclopedia/ZerosOfAnalyticFunctionsAreIsolated.html" [Broken] it follows that any [itex]0[/itex] of [itex]f[/itex] in [itex] \{ z : |z| < 1 \}[/itex] is isolated. But is it possible that the zeroes have a limit point on the boundary of convergence. For example, is possible that an analytic function with radius of convergence 1 has zeros exactly at [itex]1-\frac{1}{n}[/itex] for all [itex]n \in Z^{+}[/itex].

Take a look at the Mittag-Leffler Theorem which may be over kill. I do not really understand these issues and would be willing to go through the proofs with you.
 
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Hurkyl
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(It's hard to read that without question marks)


Can you write down an analytic function with infinitely many zeroes?

Can you apply a change of variable to move those zeroes to the unit circle?
 
  • #4
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Let me try, [tex]\frac{1}{\Gamma(z)}[/tex] is entire and has zeroes at [itex] -1,-2, \dots [/itex], so [tex]\frac{1}{\Gamma(\frac{-1}{1-z})}[/tex] would have 0's at [itex]1-\frac{1}{n}[/itex] and since the pole closest to 0 is at 1, the radius of convergence is 1.

Does that look right?

[Added later]
[itex]f(\frac{1}{1-z})[/itex] where [itex]f[/itex] is an entire function with zeroes exactly at the positive integers will do in general.
 
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