An expression in functional analysis

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Discussion Overview

The discussion revolves around the expression \(\frac{1}{z}f\left(\frac{1}{z}\right)\) in the context of functional analysis, specifically exploring theorems related to this expression. Participants are interested in theoretical implications and related mathematical concepts.

Discussion Character

  • Exploratory, Technical explanation

Main Points Raised

  • One participant inquires about theorems concerning the expression \(\frac{1}{z}f\left(\frac{1}{z}\right)\).
  • Another participant suggests a related theorem involving \(\frac{f(1/z)}{z^2}\) and the residue at infinity, stating that if \(f(w)\) has a pole at infinity, then \(\mbox{Res}_{w = \infty} f(w) = -\mbox{Res}_{z = 0} \frac{f(\frac{1}{z})}{z^2}\).
  • A later reply expresses appreciation for the provided theorem, indicating it aligns with their inquiry.
  • One participant comments on the effectiveness of vague questions in eliciting precise answers.

Areas of Agreement / Disagreement

Participants do not explicitly agree or disagree on any points, but there is a shared appreciation for the relevance of the theorem provided in response to the initial question.

Contextual Notes

The discussion does not clarify any assumptions or limitations regarding the expression or theorems mentioned, nor does it address potential dependencies on definitions or specific contexts within functional analysis.

Charles49
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Are there any theorems concerning this expression
[tex]\frac{1}{z}f\left(\frac{1}{z}\right)[/tex]. I appreciate posts of any theorems you can think of.
 
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The closest thing I can think of concerns [itex]f(1/z)/z^2[/itex]. If you have a complex function f(w) with a pole at infinity, then

[tex]\mbox{Res}_{w = \infty} f(w) = -\mbox{Res}_{z = 0} \frac{f(\frac{1}{z})}{z^2}[/tex]
 
Mute said:
The closest thing I can think of concerns [itex]f(1/z)/z^2[/itex]. If you have a complex function f(w) with a pole at infinity, then

[tex]\mbox{Res}_{w = \infty} f(w) = -\mbox{Res}_{z = 0} \frac{f(\frac{1}{z})}{z^2}[/tex]
Thanks, this was exactly what I was looking for!
 
Haha, sometimes a vague question can get the perfect answer immediately :)
 

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