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1: 22 Jacobian Elliptic Functions
Chapter 22 Jacobian Elliptic Functions
2: 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 } .
3: 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 } .
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 } .
34.7.4 ( j 13 j 23 j 33 m 13 m 23 m 33 ) { j 11 j 12 j 13 j 21 j 22 j 23 j 31 j 32 j 33 } = m r 1 , m r 2 , r = 1 , 2 , 3 ( 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 ) .
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 } .
4: William P. Reinhardt
  • In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.
    5: Peter L. Walker
    6: Staff
  • William P. Reinhardt, University of Washington, Chaps. 20, 22, 23

  • Peter L. Walker, American University of Sharjah, Chaps. 20, 22, 23

  • William P. Reinhardt, University of Washington, for Chaps. 20, 22, 23

  • Peter L. Walker, American University of Sharjah, for Chaps. 20, 22, 23

  • 7: 1.3 Determinants
    1.3.1 det [ a j k ] = | a 11 a 12 a 21 a 22 | = a 11 a 22 - a 12 a 21 .
    1.3.2 det [ a j k ] = | a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 | = a 11 | a 22 a 23 a 32 a 33 | - a 12 | a 21 a 23 a 31 a 33 | + a 13 | a 21 a 22 a 31 a 32 | = a 11 a 22 a 33 - a 11 a 23 a 32 - a 12 a 21 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 - a 13 a 22 a 31 .
    1.3.8 | a 11 a 12 a 21 a 22 | 2 ( a 11 2 + a 12 2 ) ( a 21 2 + a 22 2 ) ,
    8: 5.23 Approximations
    For additional approximations see Hart et al. (1968, Appendix B), Luke (1975, pp. 22–23), and Weniger (2003). … See Luke (1975, pp. 22–23) for additional expansions. …
    9: 26.2 Basic Definitions
    Table 26.2.1: Partitions p ( n ) .
    n p ( n ) n p ( n ) n p ( n )
    5 7 22 1002 39 31185
    8 22 25 1958 42 53174
    10: Bibliography Z
  • M. R. Zaghloul and A. N. Ali (2011) Algorithm 916: computing the Faddeyeva and Voigt functions. ACM Trans. Math. Software 38 (2), pp. Art. 15, 22.
  • M. R. Zaghloul (2017) Algorithm 985: Simple, Efficient, and Relatively Accurate Approximation for the Evaluation of the Faddeyeva Function. ACM Trans. Math. Softw. 44 (2), pp. 22:1–22:9.
  • M. I. Žurina and L. N. Karmazina (1964) Tables of the Legendre functions P - 1 / 2 + i τ ( x ) . Part I. Translated by D. E. Brown. Mathematical Tables Series, Vol. 22, Pergamon Press, Oxford.