Can some functions not be written with elementary functions?

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Discussion Overview

The discussion revolves around the question of whether certain functions, particularly those defined by integral expressions, can be expressed using elementary functions. Participants explore the nature of proofs related to this topic and its connections to Galois theory.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant inquires about the types of proofs that demonstrate some functions cannot be expressed in terms of elementary functions, expressing a lack of understanding of the underlying concepts.
  • Another participant confirms that the topic is related to Galois theory, suggesting a connection between the two areas.
  • A different participant notes their surprise at not having previously encountered the terms "differential" and "Galois" together, indicating a personal exploration of the topic.
  • One participant references Watson's Theory of Bessel Functions, suggesting that it contains extensive discussions and proofs regarding the solvability of certain differential equations in terms of elementary functions, describing the topic as tricky and tough.

Areas of Agreement / Disagreement

Participants generally agree on the connection between the topic and Galois theory, but the discussion remains unresolved regarding the specifics of the proofs and the implications for understanding functions defined by integral expressions.

Contextual Notes

The discussion does not clarify the specific assumptions or definitions that underpin the claims about functions and their expressibility in terms of elementary functions. There is also an absence of detailed mathematical steps or examples to illustrate the points raised.

jostpuur
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I've heard it is possible to prove, that some functions, for example defined with integral expressions, cannot be written with elementary functions (in somehow finite manner). Since I don't have a clue of how these things are proven, I'm now merely asking, that what kind of proofs these proofs are? What are they based on?

Since the problem setting seems similar to that of solving quintic, I must ask, does this thing have anything to do with Galois theory? I mean, could this be a reason to start studying it?

(I'm not sure wheter this belongs under analysis or algebra or logic or what...)
 
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Funny how I never happened to hit the words "differential" and "galois" into the google at the same time :rolleyes:
 
If you can find Watson's Theory of Bessel Functions, then you will run into many pages of proofs and discussions of whether certain differential equations can be solved in terms of elementary functions. Tricky and tough.

Regards,
Reilly Atkinson
 

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