
SOME HISTORICAL NOTES
From: Stephen Wolfram, A New Kind of Science Notes for Chapter 12: The Principle of Computational Equivalence
Section: Computational Irreducibility
Page 1133
Exact solutions. Some notable cases where closedform analytical results have been found in terms of standard mathematical functions include: quadratic equations (~2000 BC) (Sqrt); cubic, quartic equations (1530s) (x^(1/n)); 2body problem (1687) (Cos); catenary (1690) (Cosh); brachistochrone (1696) (Sin); spinning top (1849; 1888; 1888) (JacobiSN; WeierstrassP; hyperelliptic functions); quintic equations (1858) (EllipticTheta); halfplane diffraction (1896) (FresnelC); Mie scattering (1908) (BesselJ, BesselY, LegendreP); Einstein equations (Schwarzschild (1916), ReissnerNordström (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 (1960s1980s) (PolyLog); KdV equation (1967) (Sech etc.); Toda lattice (1967) (Sech); sixvertex spin model (1967) (Sinh integrals); CalogeroMoser model (1971) (Hypergeometric1F1); YangMills instantons (1975) (rational functions); hardhexagon spin model (1979) (EllipticTheta); additive cellular automata (1984) (MultiplicativeOrder); SeibergWitten supersymmetric theory (1994) (Hypergeometric2F1). When problems are originally stated as differential equations, results in terms of integrals ("quadrature") are sometimes considered exact solutions  as occasionally are convergent series. When one exact solution is found, there often end up being a whole family  with much investigation going into the symmetries that relate them. It is notable that when many of the examples above were discovered they were at first expected to have broad significance in their fields. But the fact that few actually did can be seen as further evidence of how narrow the scope of computational reducibility usually is. Notable examples of systems that have been much investigated, but where no exact solutions have been found include the 3D Ising model, quantum anharmonic oscillator and quantum helium atom.
Stephen Wolfram, A New Kind of Science (Wolfram Media, 2002), page 1133.
© 2002, Stephen Wolfram, LLC

