6
$\begingroup$

Let's call a real number explicit if it can be expressed starting from integers by using arithmetic operations, radicals, exponents, logarithms, trigonometric and inverse trigonometric functions. For complex numbers $i$ is allowed in addition to integers, and we get what Chow calls EL numbers. Rational numbers, $e=\exp(1)$ and $\pi=\cos^{-1}(-1)$ are explicit, some algebraic numbers and Euler's $\gamma$ probably are not. If a function has an elementary anti-derivative then a definite integral of it with explicit limits is also explicit. The converse is not true, $\int x^{-2}e^{-\frac1{x^2}}dx$ is not elementary, but $\int_0^1 x^{-2}e^{-\frac1{x^2}}dx=\frac{\sqrt{\pi}}{2}$ is explicit (this is the Gaussian integral up to substitution).

There is a theory of Liouville that allows proving that some anti-derivatives are not elementary, is there something similar for definite integrals? There was a long standing unanswered question on MSE about computing $\int_0^{\frac{\pi}{2}}\frac1{(1+x^2)(1+\tan x)}dx$, and the consensus was that it's not expressible in elementary terms. How does one prove something like that?

EDIT: Since there doesn't appear to be a general theory any non-trivial example of a proof for a particular case would be interesting. Also, explicit or EL numbers above are only an example, proving non-expressibility in some other reasonable terms would also be interesting.

$\endgroup$
6
  • $\begingroup$ No, there is not a Liouville-type theory for definite integrals. $\endgroup$
    – GEdgar
    Aug 18, 2014 at 20:52
  • $\begingroup$ Are there examples of particular cases being proved? $\endgroup$
    – Conifold
    Aug 18, 2014 at 21:15
  • $\begingroup$ Is there a single example of something (of independent interest, not constructed for this purpose) that is not an EL number? $\endgroup$
    – GEdgar
    Aug 18, 2014 at 21:23
  • $\begingroup$ Well, Chow proves assuming Schanuel's conjecture that algebraic numbers not expressible in radicals are not EL. So roots of any polynomial with unsolvable Galois group would be not EL. Actually, I'd be happy with a non-trivial example of a proof that a definite integral isn't expressible in radicals (not based on it being $\pi$ or something) or in some other recursively defined terms, EL are only an example. $\endgroup$
    – Conifold
    Aug 18, 2014 at 22:30
  • $\begingroup$ There are approaches for applying Liouville's and Risch's methods of indefinite integration in a differential field to definite integration, e.g. Raab, C. G.: Definite Integration in Differential Fields. Thesis at Johannes Kepler University Linz, 2012. $\endgroup$
    – IV_
    Nov 18, 2017 at 16:24

1 Answer 1

4
$\begingroup$

Here are a couple of excerpts that might be of use to you or others in tracking down more information about this topic:

Joseph Fels Ritt (1893-1951), Integration in Finite Terms. Liouville's Theory of Elementary Methods, Columbia University Press, 1948, ix + 100 pages.

from p. 60:

We should like to present a class of problems in which numbers are involved rather than functions.

The classification of numbers as algebraic or transcendental is well known. A number is algebraic if it satisfies an algebraic equation with integral coefficients, not all zero; otherwise the number is transcendental.

Let us define elementary number. An algebraic number will be called a number of order zero. The exponential of any algebraic number distinct from zero, or a logarithm distinct from zero of an algebraic number, will be called a monomial of order unity. A number of order unity is one which is not algebraic and satisfies an algebraic equation whose coefficients are polynomials, with integral coefficients, in monomials of order one. Continuing, we secure the elementary numbers.

There arises immediately, of course, the problem of the existence of numbers of all orders. Priority should perhaps be given to problems on the character of the roots of simple transcendental equations. One might ask, for instance, whether the equation $$e^z = z$$ has an elementary root. These are, of course, problems of greater difficulty than those which we have been studying.

Note: Ritt says nothing else about this classification of numbers in his book.

Dmitry Dmitrievich Morduhai-Boltovskoi (1875-1952), On hypertranscendental functions and hypertranscendental numbers (Russian), Doklady Akademii Nauk SSSR [= Comptes Rendus de l'Académie des Sciences de l'URSS] (N.S.) 64 (1949), 21-24.

Mathematical Reviews 10,432f, by Joseph Fels Ritt:

The author designates a function as hyperalgebraic or hypertranscendental according as it does or does not satisfy an algebraic differential equation (1) $f\left(x,\,y,\,y',\,\dots,\,y^{(n)}\right) = 0.$ If (1) has algebraic numbers for coefficients, a solution $y$ called algebraically-hyperalgebraic if $y$ and its first $n-1$ derivatives assume algebraic values for some algebraic value of $x.$ A value of such a function for any algebraic value of $x$ is called a hyperalgebraic number. Such numbers are seen to form a countable set. It is shown that if a power series represents an algebraically-hyperalgebraic function, the coefficients all lie in a field generated by an algebraic number.

Note: For an idea of what is meant by an algebraic differential equation, see my answer (and my additional comments) to the math StackExchange question Expanded concept of elementary function?.

(ADDED NEXT DAY) The following survey paper seems to be especially relevant for what you're interested in:

Periods by M. Kontsevich and D. Zagier (May 2001)

A google search for "periods" {AND} "Kontsevich" {AND} "Zagier" (which, among other things, will bring up any google-findable papers that cite Kontsevich/Zagier's survey paper) seems to lead to a lot of relevant material, although my background in this area is not sufficient to allow me to easily offer suggestions among the google-hits.

$\endgroup$
2
  • $\begingroup$ So the problem seems to be that there is no obvious way to convert values of definite integrals into equations they satisfy from the knowledge of integrands and limits. Unlike in Galois theory for roots in radicals, and in Liouville's theory for elementary anti-derivatives. $\endgroup$
    – Conifold
    Aug 19, 2014 at 20:27
  • $\begingroup$ I looked at Kontsevich-Zagier but periods are weird, $\pi$ is a period, but $\frac1{\pi}$ (conjecturally) isn't, and neither is $e$. But at least they are defined in terms of integrals so may be it's easier to connect definite integrals to them. $\endgroup$
    – Conifold
    Aug 19, 2014 at 20:31

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.