§34.8 Approximations for Large Parameters

Notes:: See Watson (1999), Chen et al. (1999).
Keywords:: $9j$ symbols, $6j$ symbols, $3j$ symbols
Permalink:: http://dlmf.nist.gov/34.8

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

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than, $\mathop{\sin\/}\nolimits z$ : sine function, $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $l,l_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.8.E1
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where

34.8.2 $\mathop{\cos\/}\nolimits\theta=\frac{j_{1}(j_{1}+1)+j_{2}(j_{2}+1)-j_{3}(j_{3}+1)}{2\sqrt{j_{1}(j_{1}+1)j_{2}(j_{2}+1)}},$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function and $j,j_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.8.E2
Encodings:: TeX, pMML, png

and the symbol $\mathop{o\/}\nolimits\!\left(1\right)$ 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 $3j$ 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.