§34.8 Approximations for Large Parameters

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

 34.8.1 $\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ j_{2}&j_{1}&l_{3}\end{Bmatrix}=(-1)^{j_{1}+j_{2}+j_{3}+l_{3}}\*\left(\frac{4}{% \pi(2j_{1}+1)(2j_{2}+1)(2l_{3}+1)\sin\theta}\right)^{\frac{1}{2}}\*\left(\cos% \left((l_{3}+\tfrac{1}{2})\theta-\tfrac{1}{4}\pi\right)+o\left(1\right)\right),$ $j_{1},j_{2},j_{3}\gg l_{3}\gg 1$,

where

 34.8.2 $\cos\theta=\frac{j_{1}(j_{1}+1)+j_{2}(j_{2}+1)-j_{3}(j_{3}+1)}{2\sqrt{j_{1}(j_% {1}+1)j_{2}(j_{2}+1)}},$ ⓘ Symbols: $\cos\NVar{z}$: cosine function and $j,j_{r}$: non-negative integers or non-negative integers plus one half. Permalink: http://dlmf.nist.gov/34.8.E2 Encodings: TeX, pMML, png See also: Annotations for 34.8 and 34

and the symbol $o\left(1\right)$ 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 $\mathit{3j}$ and $\mathit{6j}$ symbols are given in Schulten and Gordon (1975b). For approximations for the $\mathit{3j}$, $\mathit{6j}$, and $\mathit{9j}$ symbols with error bounds see Flude (1998), Chen et al. (1999), and Watson (1999): these references also cite earlier work.