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11: 34.2 Definition: 3 j Symbol
Either all of them are nonnegative integers, or one is a nonnegative integer and the other two are half-odd positive integers. …
34.2.1 | j r - j s | j t j r + j s ,
34.2.2 m r = - j r , - j r + 1 , , j r - 1 , j r , r = 1 , 2 , 3 ,
and the summation is over all nonnegative integers s such that the arguments in the factorials are nonnegative. … For alternative expressions for the 3 j symbol, written either as a finite sum or as other terminating generalized hypergeometric series F 2 3 of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).
12: 34.6 Definition: 9 j Symbol
34.6.1 { j 11 j 12 j 13 j 21 j 22 j 23 j 31 j 32 j 33 } = all  m r s ( j 11 j 12 j 13 m 11 m 12 m 13 ) ( j 21 j 22 j 23 m 21 m 22 m 23 ) ( j 31 j 32 j 33 m 31 m 32 m 33 ) ( j 11 j 21 j 31 m 11 m 21 m 31 ) ( j 12 j 22 j 32 m 12 m 22 m 32 ) ( j 13 j 23 j 33 m 13 m 23 m 33 ) ,
34.6.2 { j 11 j 12 j 13 j 21 j 22 j 23 j 31 j 32 j 33 } = j ( - 1 ) 2 j ( 2 j + 1 ) { j 11 j 21 j 31 j 32 j 33 j } { j 12 j 22 j 32 j 21 j j 23 } { j 13 j 23 j 33 j j 11 j 12 } .
The 9 j symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments. …
13: 34.7 Basic Properties: 9 j Symbol
34.7.1 { j 11 j 12 j 13 j 21 j 22 j 13 j 31 j 31 0 } = ( - 1 ) j 12 + j 21 + j 13 + j 31 ( ( 2 j 13 + 1 ) ( 2 j 31 + 1 ) ) 1 2 { j 11 j 12 j 13 j 22 j 21 j 31 } .
Odd permutations of columns or rows introduce a phase factor ( - 1 ) R , where R is the sum of all arguments of the 9 j symbol. …
34.7.2 j 12 j 34 ( 2 j 12 + 1 ) ( 2 j 34 + 1 ) ( 2 j 13 + 1 ) ( 2 j 24 + 1 ) { j 1 j 2 j 12 j 3 j 4 j 34 j 13 j 24 j } { j 1 j 2 j 12 j 3 j 4 j 34 j 13 j 24 j } = δ j 13 , j 13 δ j 24 , j 24 .
34.7.3 j 13 j 24 ( - 1 ) 2 j 2 + j 24 + j 23 - j 34 ( 2 j 13 + 1 ) ( 2 j 24 + 1 ) { j 1 j 2 j 12 j 3 j 4 j 34 j 13 j 24 j } { j 1 j 3 j 13 j 4 j 2 j 24 j 14 j 23 j } = { j 1 j 2 j 12 j 4 j 3 j 34 j 14 j 23 j } .
34.7.5 j ( 2 j + 1 ) { j 11 j 12 j j 21 j 22 j 23 j 31 j 32 j 33 } { j 11 j 12 j j 23 j 33 j } = ( - 1 ) 2 j { j 21 j 22 j 23 j 12 j j 32 } { j 31 j 32 j 33 j j 11 j 21 } .
14: 34.4 Definition: 6 j Symbol
34.4.1 { j 1 j 2 j 3 l 1 l 2 l 3 } = m r m s ( - 1 ) l 1 + m 1 + l 2 + m 2 + l 3 + m 3 ( j 1 j 2 j 3 m 1 m 2 m 3 ) ( j 1 l 2 l 3 m 1 m 2 - m 3 ) ( l 1 j 2 l 3 - m 1 m 2 m 3 ) ( l 1 l 2 j 3 m 1 - m 2 m 3 ) ,
34.4.2 { j 1 j 2 j 3 l 1 l 2 l 3 } = Δ ( j 1 j 2 j 3 ) Δ ( j 1 l 2 l 3 ) Δ ( l 1 j 2 l 3 ) Δ ( l 1 l 2 j 3 ) s ( - 1 ) s ( s + 1 ) ! ( s - j 1 - j 2 - j 3 ) ! ( s - j 1 - l 2 - l 3 ) ! ( s - l 1 - j 2 - l 3 ) ! ( s - l 1 - l 2 - j 3 ) ! 1 ( j 1 + j 2 + l 1 + l 2 - s ) ! ( j 2 + j 3 + l 2 + l 3 - s ) ! ( j 3 + j 1 + l 3 + l 1 - s ) ! ,
where the summation is over all nonnegative integers s such that the arguments in the factorials are nonnegative. …
34.4.3 { j 1 j 2 j 3 l 1 l 2 l 3 } = ( - 1 ) j 1 + j 3 + l 1 + l 3 Δ ( j 1 j 2 j 3 ) Δ ( j 2 l 1 l 3 ) ( j 1 - j 2 + l 1 + l 2 ) ! ( - j 2 + j 3 + l 2 + l 3 ) ! ( j 1 + j 3 + l 1 + l 3 + 1 ) ! Δ ( j 1 l 2 l 3 ) Δ ( j 3 l 1 l 2 ) ( j 1 - j 2 + j 3 ) ! ( - j 2 + l 1 + l 3 ) ! ( j 1 + l 2 + l 3 + 1 ) ! ( j 3 + l 1 + l 2 + 1 ) ! F 3 4 ( - j 1 + j 2 - j 3 , j 2 - l 1 - l 3 , - j 1 - l 2 - l 3 - 1 , - j 3 - l 1 - l 2 - 1 - j 1 + j 2 - l 1 - l 2 , j 2 - j 3 - l 2 - l 3 , - j 1 - j 3 - l 1 - l 3 - 1 ; 1 ) ,
For alternative expressions for the 6 j symbol, written either as a finite sum or as other terminating generalized hypergeometric series F 3 4 of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).
15: 19.23 Integral Representations
19.23.6 4 π R F ( x , y , z ) = 0 2 π 0 π sin θ d θ d ϕ ( x sin 2 θ cos 2 ϕ + y sin 2 θ sin 2 ϕ + z cos 2 θ ) 1 / 2 ,
19.23.6_5 R G ( x , y , z ) = 1 4 π 0 2 π 0 π ( x sin 2 θ cos 2 ϕ + y sin 2 θ sin 2 ϕ + z cos 2 θ ) 1 / 2 sin θ d θ d ϕ ,
19.23.9 R - a ( b ; z ) = 4 Γ ( b 1 + b 2 + b 3 ) Γ ( b 1 ) Γ ( b 2 ) Γ ( b 3 ) 0 π / 2 0 π / 2 ( j = 1 3 z j l j 2 ) - a j = 1 3 l j 2 b j - 1 sin θ d θ d ϕ , b j > 0 , z j > 0 .
19.23.10 R - a ( b ; z ) = 1 B ( a , a ) 0 1 u a - 1 ( 1 - u ) a - 1 j = 1 n ( 1 - u + u z j ) - b j d u , a , a > 0 ; a + a = j = 1 n b j ; z j ( - , 0 ] .
16: 15.19 Methods of Computation
For z it is always possible to apply one of the linear transformations in §15.8(i) in such a way that the hypergeometric function is expressed in terms of hypergeometric functions with an argument in the interval [ 0 , 1 2 ] . For z it is possible to use the linear transformations in such a way that the new arguments lie within the unit circle, except when z = e ± π i / 3 . This is because the linear transformations map the pair { e π i / 3 , e - π i / 3 } onto itself. … For example, in the half-plane z 1 2 we can use (15.12.2) or (15.12.3) to compute F ( a , b ; c + N + 1 ; z ) and F ( a , b ; c + N ; z ) , where N is a large positive integer, and then apply (15.5.18) in the backward direction. …
17: 19.16 Definitions
19.16.9 R - a ( b ; z ) = 1 B ( a , a ) 0 t a - 1 j = 1 n ( t + z j ) - b j d t = 1 B ( a , a ) 0 t a - 1 j = 1 n ( 1 + t z j ) - b j d t , b 1 + + b n > a > 0 , b j , z j ( - , 0 ] ,
19.16.12 R - a ( b 1 , , b 4 ; c - 1 , c - k 2 , c , c - α 2 ) = 2 ( sin 2 ϕ ) 1 - a B ( a , a ) 0 ϕ ( sin θ ) 2 a - 1 ( sin 2 ϕ - sin 2 θ ) a - 1 ( cos θ ) 1 - 2 b 1 ( 1 - k 2 sin 2 θ ) - b 2 ( 1 - α 2 sin 2 θ ) - b 4 d θ ,
R - a ( b ; z ) is an elliptic integral iff the z ’s are distinct and exactly four of the parameters a , a , b 1 , , b n are half-odd-integers, the rest are integers, and none of a , a , a + a is zero or a negative integer. …
18: Bibliography G
  • W. Gautschi (1965) Algorithm 259: Legendre functions for arguments larger than one. Comm. ACM 8 (8), pp. 488–492.
  • W. Gautschi (2016) Algorithm 957: evaluation of the repeated integral of the coerror function by half-range Gauss-Hermite quadrature. ACM Trans. Math. Softw. 42 (1), pp. 9:1–9:10.
  • 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.
  • A. Gil and J. Segura (1997) Evaluation of Legendre functions of argument greater than one. Comput. Phys. Comm. 105 (2-3), pp. 273–283.
  • A. Gil, J. Segura, and N. M. Temme (2004a) Algorithm 831: Modified Bessel functions of imaginary order and positive argument. ACM Trans. Math. Software 30 (2), pp. 159–164.
  • 19: Bibliography P
  • B. V. Pal tsev (1999) On two-sided estimates, uniform with respect to the real argument and index, for modified Bessel functions. Mat. Zametki 65 (5), pp. 681–692 (Russian).
  • R. B. Paris (2004) Exactification of the method of steepest descents: The Bessel functions of large order and argument. Proc. Roy. Soc. London Ser. A 460, pp. 2737–2759.
  • S. Paszkowski (1991) Evaluation of the Fermi-Dirac integral of half-integer order. Zastos. Mat. 21 (2), pp. 289–301.
  • 20: 4.37 Inverse Hyperbolic Functions
    Graphs of the principal values for real arguments are given in §4.29. This section also indicates conformal mappings, and surface plots for complex arguments. …
    4.37.22 arccosh x = ± ln ( i ( 1 - x 2 ) 1 / 2 + x ) , x ( - 1 , 1 ] ,