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From: Stephen Wolfram, A New Kind of Science
Notes for Chapter 10: Processes of Perception and Analysis
Section: Traditional Mathematics and Mathematical Formulas
Page 1091

Standard mathematical functions. There are an infinite number of possible functions with integer or continuous arguments. But in practice there is a definite set of standard named mathematical functions that are considered reasonable to include as primitives in formulas, and that are implemented as built-in functions in Mathematica. The so-called elementary functions (logarithms, exponentials, trigonometric and hyperbolic functions, and their inverses) were mostly introduced before about 1700. In the 1700s and 1800s another several hundred so-called special functions were introduced. Most arose first as solutions to specific differential equations, typically in physics and astronomy; some arose as products, sums of series or inverses of other functions. In the mid-1800s it became clear that despite their different origins most of these functions could be viewed as special cases of Hypergeometric2F1[a, b, c, z], and that the functions covered the solutions to all linear differential equations of a certain type. (Zeta and PolyLog are parametric derivatives of Hypergeometric2F1; elliptic modular functions are inverses.) Rather few new special functions have been introduced over the past century. The main reason has been that the obvious generalizations seem to yield classes of functions whose properties cannot be worked out with much completeness. So, for example, if there are more parameters it becomes difficult to find continuous definitions that work for all complex values of these parameters. (Typically one needs to generalize formulas that are initially set up with integer numbers of terms; examples include taking Power[x, y] to be Exp[Log[x] y] and x! to be Gamma[x+1].) And if one modifies the usual hypergeometric equation y’’[x]==f[y[x], y’[x]] by making f nonlinear then solutions typically become hard to find, and vary greatly in character with the form of f. (For rational f Paul Painlevé in the 1890s identified just 6 additional types of functions that are needed, but even now series expansions are not known for all of them.) Generalizations of special functions can in principle be used to represent the results of many kinds of computations. Thus, for example, generalized elliptic theta functions represent solutions to arbitrary polynomial equations, while multivariate hypergeometric functions represent arbitrary conformal mappings. In Mathematica, however, functions like Root provide more convenient ways to access such results.

A variety of standard mathematical functions with integer arguments were introduced in the late 1800s and early 1900s in connection with number theory. A few functions that involve manipulation of digits have also become standard since the use of computers became widespread.

Stephen Wolfram, A New Kind of Science (Wolfram Media, 2002), page 1091.
© 2002, Stephen Wolfram, LLC