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Prove or disprove: A non-constant analytic function can have at most finitely many zeros on a
closed disk.
Solution
Suppose an analytic function f has infinitely many zeros on some closed disk D. Then there
exists a sequence of zeros in D with a limit point in D. Thus by the identity theorem (Let D be a
domain and f analytic in D. If the set of zeros Z(f) has a limit point in D, then f 0 in D.), f is
identically zero and thus constant.
so for a non-constant analytic function must have finite zeros on a closed disk

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Prove or disprove A non-constant analytic function can have at most .pdf

  • 1. Prove or disprove: A non-constant analytic function can have at most finitely many zeros on a closed disk. Solution Suppose an analytic function f has infinitely many zeros on some closed disk D. Then there exists a sequence of zeros in D with a limit point in D. Thus by the identity theorem (Let D be a domain and f analytic in D. If the set of zeros Z(f) has a limit point in D, then f 0 in D.), f is identically zero and thus constant. so for a non-constant analytic function must have finite zeros on a closed disk