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34 3j,6j,9j SymbolsProperties

§34.8 Approximations for Large Parameters

For large values of the parameters in the 3j, 6j, and 9j symbols, different asymptotic forms are obtained depending on which parameters are large. For example,

34.8.1 {j1j2j3j2j1l3}=(-1)j1+j2+j3+l3(4π(2j1+1)(2j2+1)(2l3+1)sinθ)12(cos((l3+12)θ-14π)+o(1)),
j1,j2,j3l31,

where

34.8.2 cosθ=j1(j1+1)+j2(j2+1)-j3(j3+1)2j1(j1+1)j2(j2+1),

and the symbol o(1) denotes a quantity that tends to zero as the parameters tend to infinity, as in §2.1(i).

Semiclassical (WKBJ) approximations in terms of trigonometric or exponential functions are given in Varshalovich et al. (1988, §§8.9, 9.9, 10.7). Uniform approximations in terms of Airy functions for the 3j and 6j symbols are given in Schulten and Gordon (1975b). For approximations for the 3j, 6j, and 9j symbols with error bounds see Flude (1998), Chen et al. (1999), and Watson (1999): these references also cite earlier work.