Derivations of the series expansions

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Discussion Overview

The discussion revolves around the derivations of Taylor, Fourier, and Laurent series, exploring the foundational concepts and historical context behind these mathematical constructs. Participants express interest in understanding how these series are originally derived rather than just their applications or coefficient calculations.

Discussion Character

  • Exploratory, Technical explanation, Historical

Main Points Raised

  • One participant inquires about the derivations of Taylor, Fourier, and Laurent series specifically.
  • Another participant acknowledges that these series did not emerge without basis and asks for clarification on the inquiry.
  • A participant mentions that while they understand how to derive coefficients from the series forms, they are uncertain about the original derivation of the series themselves.
  • One reply suggests that the concept of 'inspiration' plays a role in the derivation process and references historical developments in mathematics related to infinite series.
  • Links to Wikipedia articles on Fourier, Taylor, and Laurent series are provided for further reading.
  • Historical context is introduced, noting that the convergence of infinite series was established in the 17th century, which allowed for their application in various mathematical areas.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the derivation of these series, with some uncertainty about the foundational aspects. No consensus is reached on the original derivation methods.

Contextual Notes

The discussion highlights a lack of clarity on the initial derivation processes for the series, with participants relying on established forms and transformations without fully addressing the foundational derivations.

Benjam:n
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Are there derivations of the taylor, Fourier and laurant series?
 
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Of course. They didn't just fall out of the sky. Specifically what do you mean?
 
In all three cases I know how if you accept that they can be written in that form (I.e. as power series or infinite series of the sines and cosines), then you can derive the coefficients using cleverly picked transformations, i.e. differentiation, the Fourier transforms or Cauchy integral formula trick. What I don't know is how you derive the original bit.
 

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