Chapter 4: Systems Based on Numbers

Section 6: Mathematical Functions

Mathematical functions

(See page 1091.) BesselJ[0,x] goes like Sin[x]/Sqrt[x] for large x while AiryAi[-x] goes like Sin[x^(3/2)]/x^(1/4). Other standard mathematical functions that oscillate at large x include JacobiSN and MathieuC. Most hypergeometric-type functions either increase or decrease exponentially for large arguments, though in the directions of Stokes lines in the complex plane they can oscillate sinusoidally. (For AiryAi[x] the Stokes lines are in directions (-1)^({1,2,3}/3).)

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