Considering that transcendental numbers are described as not a root of a non-zero polynomial equation with rational coefficients, are there classifications of transcendental numbers that are considered not algebraic but the solution to a definite integral of an algebraic function with algebraic coefficients and limits of integration? We can get the transcendental number $\pi$ from such an expression, but perhaps there are other computable numbers that can't be extracted from such an operation.

That is, is there a notion of transcendence theory that studies independence of functions (or differential equations) in the same way that we study algebraic independence of numbers?

Further generalizing, I was motivated by the categorization of numbers between integral, rational, algebraic, transcendental, and uncomputable, and it occurred to me that there seems like there ought to be different classifications of transcendental numbers between non-algebraic and uncomputable.

Does this have any relation to differential Galois theory?

  • $\begingroup$ Just a minor comment: It seems to me that such "integrally defined numbers" are still countably infinite. So they have measure zero in the real number line. $\endgroup$ – Michael Jul 4 '14 at 21:49
  • $\begingroup$ I agree that $\pi$ is an integrally defined number, as is the natural log of any positive integer. I cannot figure out if $e$ is, or not. $\endgroup$ – Michael Jul 4 '14 at 22:00
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    $\begingroup$ See ring of periods. $\endgroup$ – Lucian Jul 4 '14 at 22:13
  • $\begingroup$ @Lucian : Good link, seems like "period" is exactly the thing. In reading the wikipedia page, it seems that nobody yet knows if $e$ works. $\endgroup$ – Michael Jul 4 '14 at 22:20
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    $\begingroup$ Also of possible interest: Closed forms: What they are and why they matter by Jonathan M. Borwein and Richard E. Crandell (and pp. 265-293 here). $\endgroup$ – Dave L. Renfro Jul 10 '14 at 17:29

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