Can some functions not be written with elementary functions?

In summary, the conversation discusses the possibility of proving that some functions, specifically those defined with integral expressions, cannot be written in a finite manner using elementary functions. The conversation also suggests a connection to Galois theory and the question of whether studying it may be helpful in understanding these proofs. It is mentioned that this topic could fall under the categories of analysis, algebra, or logic. The discussion also references Watson's Theory of Bessel Functions as a source for further exploration of this topic.
  • #1
jostpuur
2,116
19
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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  • #3
Funny how I never happened to hit the words "differential" and "galois" into the google at the same time :rolleyes:
 
  • #4
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
 

1. What are "impossible expressions" in science?

"Impossible expressions" in science refer to mathematical or physical equations or concepts that cannot be solved or achieved within the constraints of our current understanding of the universe.

2. How are impossible expressions used in science?

Impossible expressions are often used as thought experiments to push the boundaries of scientific understanding and explore the limits of our current knowledge. They can also be used to identify areas where further research and advancements are needed.

3. Can impossible expressions ever be solved?

In some cases, with new discoveries and advancements in science, what was once considered an impossible expression may eventually be solvable. However, there are also instances where some expressions may remain impossible to solve due to the fundamental laws of nature.

4. What is the significance of studying impossible expressions?

Studying impossible expressions can lead to new insights and discoveries in science, as well as inspire new ways of thinking and problem-solving. It also highlights the limits of our current understanding and motivates further research to overcome these limitations.

5. Can impossible expressions have practical applications?

While the primary purpose of studying impossible expressions is to advance scientific knowledge, there have been instances where seemingly impossible concepts have been applied in practical ways. For example, the concept of teleportation, once thought to be impossible, is now being explored and developed for potential use in quantum computing.

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