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

\mathit{9j}

§34.8 Approximations for Large Parameters

ⓘ

Keywords:: approximations for large parameters, $\mathit{9j}$ 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 Ch.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		$\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ j_{2}&j_{1}&l_{3}\end{Bmatrix}=(-1)^{j_{1}+j_{2}+j_{3}+l_{3}}\\left(\frac{4}{% \pi(2j_{1}+1)(2j_{2}+1)(2l_{3}+1)\sin\theta}\right)^{\frac{1}{2}}\\left(\cos% \left((l_{3}+\tfrac{1}{2})\theta-\tfrac{1}{4}\pi\right)+o\left(1\right)\right),$
$j_{1},j_{2},j_{3}\gg l_{3}\gg 1$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $o\left(\NVar{x}\right)$ : order less than, $\sin\NVar{z}$ : 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, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half. Permalink: http://dlmf.nist.gov/34.8.E1 Encodings: TeX, pMML, png See also: Annotations for §34.8 and Ch.34

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 Ch.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.

34.7 Basic Properties:

\mathit{9j}

Symbol 34.9 Graphical Method

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