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1: DLMF Project News
error generating summary
2: Bibliography M
  • K. L. Majumder and G. P. Bhattacharjee (1973) Algorithm AS 63. The incomplete beta integral. Appl. Statist. 22 (3), pp. 409–411.
  • J. P. McClure and R. Wong (1978) Explicit error terms for asymptotic expansions of Stieltjes transforms. J. Inst. Math. Appl. 22 (2), pp. 129–145.
  • R. Milson (2017) Exceptional orthogonal polynomials.
  • P. M. Morse and H. Feshbach (1953a) Methods of Theoretical Physics. Vol. 1, McGraw-Hill Book Co., New York.
  • P. M. Morse and H. Feshbach (1953b) Methods of Theoretical Physics. Vol. 2, McGraw-Hill Book Co., New York.
  • 3: 22 Jacobian Elliptic Functions
    Chapter 22 Jacobian Elliptic Functions
    4: 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 } .
    5: 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 } .
    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

  • Hans Volkmer, University of Wisconsin–Milwaukee, for Chaps. 22, 28, 29, 30, 31

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

  • 7: William P. Reinhardt
  • In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.
    8: Hans Volkmer
    9: Peter L. Walker
    10: 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. …