Holomorphic function on the unit disc

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Discussion Overview

The discussion revolves around the existence of a holomorphic function f(z) defined on the unit disc that satisfies the conditions f(1/n) = f(-1/n) = 1/n^3 for every natural number n. The scope includes theoretical exploration of holomorphic functions and continuity.

Discussion Character

  • Debate/contested
  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant questions the existence of a holomorphic function f(z) that meets the specified conditions.
  • Another participant asserts that there does not exist even a continuous function that satisfies these conditions.
  • A different participant suggests constructing a function g such that g(1/n) = g(-1/n) = 1/n^2 and considers the ratio f/g to prove the claim, but expresses difficulty in proceeding.
  • There is a query about the value of f(0), indicating a consideration of the function's behavior at that point.
  • One participant references the continuity of the function z \mapsto |z| and challenges the feasibility of the original claim by noting that |1/n| does not equal 1/n^3.
  • Another participant suggests examining the zeros of f(z) - z^3 to determine if they form an isolated set of points, implying a potential avenue for exploration.
  • Participants acknowledge misunderstandings about continuity, with one correcting their earlier statement regarding the function's continuity.

Areas of Agreement / Disagreement

Participants generally disagree on the existence of such a holomorphic function, with some asserting that no continuous function can satisfy the conditions, while others propose methods to explore the problem further.

Contextual Notes

There are unresolved assumptions regarding the properties of holomorphic functions and the implications of continuity in this context. The discussion also reflects varying interpretations of the mathematical conditions presented.

iamqsqsqs
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Does there exist a holomorphic function f(z) on the unit disc and satisfies f(1/n) = f(-1/n) = 1/n^3 for every n in N?
 
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There does not even exist a continuous function that does this.
 
How can we vigorously prove that? I am thinking of construct a function g such that g(1/n) = g(-1/n) = 1/n^2 and consider f/g to do it. However I am stuck and cannot go on
 
What would f(0) be?
 
micromass said:
There does not even exist a continuous function that does this.

Last time I checked, [itex]z \mapsto |z|[/itex] was continuous...

iamqsqsqs, try looking at the zeros of f(z) - z^3. Do they form an isolated set of points?
 
Citan Uzuki said:
Last time I checked [itex]z \mapsto |z|[/itex] was continuous...


*** Last time I checked [itex]\,\,\displaystyle{\left|\frac{1}{n}\right|\neq \frac{1}{n^3}}[/itex] ...

DonAntonio ***


iamqsqsqs, try looking at the zeros of f(z) - z^3. Do they form an isolated set of points?

...
 
Sorry, brain fart. I meant to say [itex]z \mapsto |z|^3[/itex]
 
Citan Uzuki said:
Sorry, brain fart. I meant to say [itex]z \mapsto |z|^3[/itex]


Hehe...yes, I supposed so. Happens to me all the time. Your answer to look at the zeroes of [itex]\,\,f(z)-z^3\,\,[/itex] pretty much wraps this up, though.

DonAntonio
 

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