Notes

Section 5: Mathematical Constants

Number classification

One can imagine classifying real numbers in terms of what kinds of operations are needed to obtain them from integers. Rational numbers require only division (or solving linear equations), while algebraic numbers require solving polynomial equations. Rather little is known about numbers that require solving transcendental equations—and indeed it can even be undecidable (see page 1138) whether two equations can yield the same number. Starting with integers and then applying arithmetic operations and fractional powers one can readily reproduce all algebraic numbers up to degree 4, but not beyond. The sets of numbers that can be obtained by applying elementary functions like Exp, Log and Sin seem in various ways to be disjoint from algebraic numbers. But if one applies multivariate elliptic or hypergeometric functions it was established in the late 1800s and early 1900s that one can in principle reach any algebraic number. One can also ask what numbers can be generated by integrals (or by solving differential equations). For rational functions f[x], Integrate[f[x], {x,0,1}] must always be a linear function of Log and ArcTan applied to algebraic numbers (f[x]=1/(1+x2) for example yields Pi/4). Multiple integrals of rational functions can be more complicated, as in

Integrate[1/(1+x2+y2),{x,0,1},{y,0,1}]== HypergeometricPFQ[{1/2, 1, 1}, {3/2, 3/2}, 1/9]/6 + Pi ArcSinh[1]/2 - Catalan

and presumably often cannot be expressed at all in terms of standard mathematical functions. Integrals of rational functions over regions defined by polynomial inequalities have recently been discussed under the name "periods". Many numbers associated with Zeta and Gamma can readily be generated, though apparently for example E and EulerGamma cannot. One can also consider numbers obtained from infinite sums (or by solving recurrence equations). If f[n] is a rational function, Sum[f[n], {n, Infinity}] must just be a linear combination of PolyGamma functions, but again the multivariate case can be much more complicated.

From Stephen Wolfram: A New Kind of Science [citation]