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As an example, the n th digit of Log[2] in base 2 is formally given by Round[FractionalPart[2 n Sum[2 -k /k, {k, ∞ }]]] . … The same basic approach as for Log[2] can be used to obtain base 16 digits in π from the following formula for π : Sum[16 -k (4/(8k + 1) - 2/(8k + 4) - 1/(8k + 5)-1/(8k + 6)), {k, 0, ∞ }] A similar approach can also be used for many other constants that can be viewed as related to values of PolyLog .
Ignoring parts that depend on particle masses the result (derived in successive orders from 1, 1, 7, 72, 891 diagrams) is 2 ( 1 + α /(2 π ) + (3 Zeta[3]/4 - 1/2 π 2 Log[2] + π 2 /12 + 197/144) ( α / π ) 2 + (83/72 π 2 Zeta[3] - 215 Zeta[5]/24 - 239 π 4 /2160 + 139 Zeta[3]/18 + 25/18 (24 PolyLog[ 4, 1/2] + Log[2] 4 - π 2 Log[2] 2 ) - 298/9 π 2 Log[2] + 17101 π 2 /810 + 28259/5184) ( α / π ) 3 - 1.4 ( α / π ) 4 + …), or roughly 2. + 0.32 α - 0.067 α 2 + 0.076 α 3 - 0.029 α 4 + … The comparative simplicity of the symbolic forms here (which might get still simpler in terms of suitable generalized polylogarithm functions) may be a hint that methods much more efficient than explicit Feynman diagram evaluation could be used.
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.) … (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!
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. … 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 + x 2 ) for example yields π /4 ). … If f[n] is a rational function, Sum[f[n], {n, ∞ }] must just be a linear combination of PolyGamma functions, but again the multivariate case can be much more complicated.
[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 ).
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