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1: 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 } .
2: Staff
  • Richard B. Paris, University of Abertay, Chaps. 8, 11

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

  • Hans Volkmer, University of Wisconsin, Milwaukee, Chaps. 29, 30

  • Richard B. Paris, University of Abertay Dundee, for Chaps. 8, 11

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

  • 3: 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
    6 11 23 1255 40 37338
    9 30 26 2436 43 63261
    11 56 28 3718 45 89134
    13 101 30 5604 47 1 24754
    4: 24.2 Definitions and Generating Functions
    Table 24.2.3: Bernoulli numbers B n = N / D .
    n N D B n
    30 861 58412 76005 14322 6.01580 8739 ×10⁸
    Table 24.2.4: Euler numbers E n .
    n E n
    30 44 15438 93249 02310 45536 82821
    Table 24.2.5: Coefficients b n , k of the Bernoulli polynomials B n ( x ) = k = 0 n b n , k x k .
    k
    n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
    11 0 5 6 0 11 2 0 11 0 11 0 55 6 11 2 1
    12 691 2730 0 5 0 33 2 0 22 0 33 2 0 11 6 1
    5: 26.9 Integer Partitions: Restricted Number and Part Size
    Table 26.9.1: Partitions p k ( n ) .
    n k
    6 0 1 4 7 9 10 11 11 11 11 11
    7 0 1 4 8 11 13 14 15 15 15 15
    8 0 1 5 10 15 18 20 21 22 22 22
    9 0 1 5 12 18 23 26 28 29 30 30
    10 0 1 6 14 23 30 35 38 40 41 42
    6: 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 } .
    7: 11 Struve and Related Functions
    Chapter 11 Struve and Related Functions
    8: 13.30 Tables
  • Slater (1960) tabulates M ( a , b , x ) for a = 1 ( .1 ) 1 , b = 0.1 ( .1 ) 1 , and x = 0.1 ( .1 ) 10 , 7–9S; M ( a , b , 1 ) for a = 11 ( .2 ) 2 and b = 4 ( .2 ) 1 , 7D; the smallest positive x -zero of M ( a , b , x ) for a = 4 ( .1 ) 0.1 and b = 0.1 ( .1 ) 2.5 , 7D.

  • Zhang and Jin (1996, pp. 411–423) tabulates M ( a , b , x ) and U ( a , b , x ) for a = 5 ( .5 ) 5 , b = 0.5 ( .5 ) 5 , and x = 0.1 , 1 , 5 , 10 , 20 , 30 , 8S (for M ( a , b , x ) ) and 7S (for U ( a , b , x ) ).

  • 9: 27.2 Functions
    Table 27.2.1: Primes.
    n p n p n + 10 p n + 20 p n + 30 p n + 40 p n + 50 p n + 60 p n + 70 p n + 80 p n + 90
    5 11 47 97 149 197 257 313 379 439 499
    Table 27.2.2: Functions related to division.
    n ϕ ( n ) d ( n ) σ ( n ) n ϕ ( n ) d ( n ) σ ( n ) n ϕ ( n ) d ( n ) σ ( n ) n ϕ ( n ) d ( n ) σ ( n )
    3 2 2 4 16 8 5 31 29 28 2 30 42 12 8 96
    4 2 3 7 17 16 2 18 30 8 8 72 43 42 2 44
    5 4 2 6 18 6 6 39 31 30 2 32 44 20 6 84
    11 10 2 12 24 8 8 60 37 36 2 38 50 20 6 93
    10: Publications
  • B. V. Saunders and Q. Wang (2005) Boundary/Contour Fitted Grid Generation for Effective Visualizations in a Digital Library of Mathematical Functions, Proceedings of the 9th International Conference on Numerical Grid Generation in Computational Field Simulations, San Jose, June 11–18, 2005. pp. 61–71. PDF
  • Q. Wang and B. V. Saunders (2005) Web-Based 3D Visualization in a Digital Library of Mathematical Functions, Proceedings of the Web3D Symposium, Bangor, UK, March 29–April 1, 2005. PDF
  • A. Youssef (2007) Methods of Relevance Ranking and Hit-content Generation in Math Search, Proceedings of Mathematical Knowledge Management (MKM2007), RISC, Hagenberg, Austria, June 27–30, 2007. PDF