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1: 1.9 Calculus of a Complex Variable
Triangle Inequality
2: 34.4 Definition: 6 j Symbol
Except in degenerate cases the combination of the triangle inequalities for the four 3 j symbols in (34.4.1) is equivalent to the existence of a tetrahedron (possibly degenerate) with edges of lengths j 1 , j 2 , j 3 , l 1 , l 2 , l 3 ; see Figure 34.4.1. …
3: Bibliography G
  • A. Gervois and H. Navelet (1984) Some integrals involving three Bessel functions when their arguments satisfy the triangle inequalities. J. Math. Phys. 25 (11), pp. 3350–3356.
  • 4: 34.5 Basic Properties: 6 j Symbol
    In the following equations it is assumed that the triangle inequalities are satisfied and that J is again defined by (34.3.4). …
    5: 1.2 Elementary Algebra
    The triangle inequality, …
    6: 28.29 Definitions and Basic Properties
    For a given ν , the characteristic equation ( λ ) 2 cos ( π ν ) = 0 has infinitely many roots λ . Conversely, for a given λ , the value of ( λ ) is needed for the computation of ν . …
    28.29.16 λ n , n = 0 , 1 , 2 , ,  with  ( λ n ) = 2 ,
    28.29.17 μ n , n = 1 , 2 , 3 , ,  with  ( μ n ) = 2 .
    Both λ n and μ n as n , and interlace according to the inequalities
    7: 23.22 Methods of Computation
  • (a)

    In the general case, given by c d 0 , we compute the roots α , β , γ , say, of the cubic equation 4 t 3 c t d = 0 ; see §1.11(iii). These roots are necessarily distinct and represent e 1 , e 2 , e 3 in some order.

    If c and d are real, and the discriminant is positive, that is c 3 27 d 2 > 0 , then e 1 , e 2 , e 3 can be identified via (23.5.1), and k 2 , k 2 obtained from (23.6.16).

    If c 3 27 d 2 < 0 , or c and d are not both real, then we label α , β , γ so that the triangle with vertices α , β , γ is positively oriented and [ α , γ ] is its longest side (chosen arbitrarily if there is more than one). In particular, if α , β , γ are collinear, then we label them so that β is on the line segment ( α , γ ) . In consequence, k 2 = ( β γ ) / ( α γ ) , k 2 = ( α β ) / ( α γ ) satisfy k 2 0 k 2 (with strict inequality unless α , β , γ are collinear); also | k 2 | , | k 2 | 1 .

    Finally, on taking the principal square roots of k 2 and k 2 we obtain values for k and k that lie in the 1st and 4th quadrants, respectively, and 2 ω 1 , 2 ω 3 are given by

    23.22.1 2 ω 1 M ( 1 , k ) = 2 i ω 3 M ( 1 , k ) = π 3 c ( 2 + k 2 k 2 ) ( k 2 k 2 ) d ( 1 k 2 k 2 ) ,

    where M denotes the arithmetic-geometric mean (see §§19.8(i) and 22.20(ii)). This process yields 2 possible pairs ( 2 ω 1 , 2 ω 3 ), corresponding to the 2 possible choices of the square root.