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For the Klein–Gordon equation, however, there is an exact solution:
u[t, x] = If[x 2 > t 2 , 0, BesselJ[0, Sqrt[t 2 - x 2 ]]]

BesselJ[0, x] goes like Sin[x]/ √ x for large x while AiryAi[-x] goes like Sin[x 3/2 ]/x 1/4 .

[History of] exact solutions
Some notable cases where closed-form analytical results have been found in terms of standard mathematical functions include: quadratic equations (~2000 BC) ( Sqrt ); cubic, quartic equations (1530s) ( x 1/n ); 2-body problem (1687) ( Cos ); catenary (1690) ( Cosh ); brachistochrone (1696) ( Sin ); spinning top (1849; 1888; 1888) ( JacobiSN ; WeierstrassP ; hyperelliptic functions); quintic equations (1858) ( EllipticTheta ); half-plane diffraction (1896) ( FresnelC ); Mie scattering (1908) ( BesselJ , BesselY , LegendreP ); Einstein equations (Schwarzschild (1916), Reissner–Nordström (1916), Kerr (1963) solutions) (rational and trigonometric functions); quantum hydrogen atom and harmonic oscillator (1927) ( LaguerreL , HermiteH ); 2D Ising model (1944) ( Sinh , EllipticK ); various Feynman diagrams (1960s-1980s) ( PolyLog ); KdV equation (1967) ( Sech etc.); Toda lattice (1967) ( Sech ); six-vertex spin model (1967) ( Sinh integrals); Calogero–Moser model (1971) ( Hypergeometric1F1 ); Yang–Mills instantons (1975) (rational functions); hard-hexagon spin model (1979) ( EllipticTheta ); additive cellular automata (1984) ( MultiplicativeOrder ); Seiberg–Witten supersymmetric theory (1994) ( Hypergeometric2F1 ).

It is also known for example that Gamma[1/3] and BesselJ[0, n] are transcendental.

The sequence of odd numbers gives the continued fraction for Coth[1] ; the sequence of even numbers for BesselI[0, 1]/BesselI[1, 1] . In general, continued fractions whose n th term is a n + b correspond to numbers given by BesselI[b/a, 2/a]/BesselI[b/a + 1, 2/a] . … (This particular example can be understood from the fact that as d increases Exp[ π √ d ] becomes extremely close to -1728 KleinInvariantJ[(1 + √ -d )/2] , which turns out to be an integer whenever there is unique factorization of numbers of the form a + b √ -d —and d = 163 is the largest of the 9 cases for which this is so.)