Is f(x) Holomorphic if f(x)^2 and f(x)^3 Are?

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Discussion Overview

The discussion revolves around the question of whether a function f(x) is holomorphic if its squares and cubes, f(x)^2 and f(x)^3, are known to be holomorphic. Participants explore potential methods of proof and the implications of singularities.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions if f(x) can be concluded as holomorphic based solely on the holomorphic nature of f(x)^2 and f(x)^3, seeking a proof.
  • Another participant suggests dividing f(x)^2 by f(x)^3 as a potential method to investigate the holomorphic nature of f(x).
  • A different participant emphasizes the need to prove that singularities must be isolated, implying that if they are not, f(z) would be constantly zero.
  • One participant reiterates the initial question about proving the holomorphic nature of f(x) and mentions the property that the derivatives of a holomorphic function are also holomorphic.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether f(x) is holomorphic based on the conditions provided. Multiple approaches and viewpoints are presented, indicating an ongoing debate.

Contextual Notes

The discussion includes assumptions about the nature of singularities and the properties of holomorphic functions, which remain unresolved.

iamqsqsqs
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If we know f(x)^2 and f(x)^3 are both holomorphic, can we say that f(x) itself is also holomorphic? And how to prove that?
 
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have you tried dividing them?
 
just figure out how to do it. just divide them and prove that the singularities must be isolated. otherwise f(z) will be constantly 0.
 
iamqsqsqs said:
If we know f(x)^2 and f(x)^3 are both holomorphic, can we say that f(x) itself is also holomorphic? And how to prove that?

prove that the derivatives of a holomorphic function are holomorphic
 

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