34 3j, 6j, 9j Symbols Properties 34.7 Basic Properties:

\mathit{9j}

§34.8 Approximations for Large Parameters

Keywords:: $\mathit{9j}$ symbols, $\mathit{6j}$ symbols, $\mathit{3j}$ symbols
Notes:: See Watson (1999), Chen et al. (1999).
Permalink:: http://dlmf.nist.gov/34.8
See also:: Annotations for 34

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		$\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ j_{2}&j_{1}&l_{3}\end{Bmatrix}=(-1)^{j_{1}+j_{2}+j_{3}+l_{3}}\\left(\frac{4}{% \pi(2j_{1}+1)(2j_{2}+1)(2l_{3}+1)\mathop{\sin\/}\nolimits\theta}\right)^{\frac% {1}{2}}\\left(\mathop{\cos\/}\nolimits\left((l_{3}+\tfrac{1}{2})\theta-\tfrac% {1}{4}\pi\right)+\mathop{o\/}\nolimits\!\left(1\right)\right),$
$j_{1},j_{2},j_{3}\gg l_{3}\gg 1$ ,
Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathop{\cos\/}\nolimits\NVar{z}$ : cosine function, $\mathop{o\/}\nolimits\!\left(\NVar{x}\right)$ : order less than, $\mathop{\sin\/}\nolimits\NVar{z}$ : sine function, $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : nonnegative integer and $l,l_{r}$ : nonnegative integer Permalink: http://dlmf.nist.gov/34.8.E1 Encodings: TeX, pMML, png See also: Annotations for 34.8

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

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 $\mathit{3j}$ 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.

\mathit{9j}

Symbol 34.9 Graphical Method