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1: Publications
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  • B. I. Schneider, B. R. Miller and B. V. Saunders (2018) NIST’s Digital Library of Mathematial Functions, Physics Today 71, 2, 48 (2018), pp. 48–53. PDF
  • 2: 9.9 Zeros
    β–ΊThey are denoted by a k , a k , b k , b k , respectively, arranged in ascending order of absolute value for k = 1 , 2 , . β–ΊIf k is regarded as a continuous variable, then … β–ΊFor large k β–Ί
    9.9.6 a k = T ⁑ ( 3 8 ⁒ Ο€ ⁒ ( 4 ⁒ k 1 ) ) ,
    β–ΊFor error bounds for the asymptotic expansions of a k , b k , a k , and b k see Pittaluga and Sacripante (1991), and a conjecture given in Fabijonas and Olver (1999). …
    3: Bibliography N
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  • W. J. Nellis and B. C. Carlson (1966) Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.
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  • E. W. Ng and M. Geller (1969) A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
  • 4: 28.35 Tables
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  • Ince (1932) includes eigenvalues a n , b n , and Fourier coefficients for n = 0 or 1 ⁒ ( 1 ) ⁒ 6 , q = 0 ⁒ ( 1 ) ⁒ 10 ⁒ ( 2 ) ⁒ 20 ⁒ ( 4 ) ⁒ 40 ; 7D. Also ce n ⁑ ( x , q ) , se n ⁑ ( x , q ) for q = 0 ⁒ ( 1 ) ⁒ 10 , x = 1 ⁒ ( 1 ) ⁒ 90 , corresponding to the eigenvalues in the tables; 5D. Notation: a n = 𝑏𝑒 n 2 ⁒ q , b n = π‘π‘œ n 2 ⁒ q .

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  • National Bureau of Standards (1967) includes the eigenvalues a n ⁑ ( q ) , b n ⁑ ( q ) for n = 0 ⁒ ( 1 ) ⁒ 3 with q = 0 ⁒ ( .2 ) ⁒ 20 ⁒ ( .5 ) ⁒ 37 ⁒ ( 1 ) ⁒ 100 , and n = 4 ⁒ ( 1 ) ⁒ 15 with q = 0 ⁒ ( 2 ) ⁒ 100 ; Fourier coefficients for ce n ⁑ ( x , q ) and se n ⁑ ( x , q ) for n = 0 ⁒ ( 1 ) ⁒ 15 , n = 1 ⁒ ( 1 ) ⁒ 15 , respectively, and various values of q in the interval [ 0 , 100 ] ; joining factors g e , n ⁑ ( q ) , f e , n ⁑ ( q ) for n = 0 ⁒ ( 1 ) ⁒ 15 with q = 0 ⁒ ( .5 ⁒  to  ⁒ 10 ) ⁒ 100 (but in a different notation). Also, eigenvalues for large values of q . Precision is generally 8D.

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  • Zhang and Jin (1996, pp. 521–532) includes the eigenvalues a n ⁑ ( q ) , b n + 1 ⁑ ( q ) for n = 0 ⁒ ( 1 ) ⁒ 4 , q = 0 ⁒ ( 1 ) ⁒ 50 ; n = 0 ⁒ ( 1 ) ⁒ 20 ( a ’s) or 19 ( b ’s), q = 1 , 3 , 5 , 10 , 15 , 25 , 50 ⁒ ( 50 ) ⁒ 200 . Fourier coefficients for ce n ⁑ ( x , 10 ) , se n + 1 ⁑ ( x , 10 ) , n = 0 ⁒ ( 1 ) ⁒ 7 . Mathieu functions ce n ⁑ ( x , 10 ) , se n + 1 ⁑ ( x , 10 ) , and their first x -derivatives for n = 0 ⁒ ( 1 ) ⁒ 4 , x = 0 ⁒ ( 5 ∘ ) ⁒ 90 ∘ . Modified Mathieu functions Mc n ( j ) ⁑ ( x , 10 ) , Ms n + 1 ( j ) ⁑ ( x , 10 ) , and their first x -derivatives for n = 0 ⁒ ( 1 ) ⁒ 4 , j = 1 , 2 , x = 0 ⁒ ( .2 ) ⁒ 4 . Precision is mostly 9S.

  • 5: 8 Incomplete Gamma and Related
    Functions
    6: 13.30 Tables
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  • 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 ) ).

  • 7: 9.18 Tables
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  • Miller (1946) tabulates a k , Ai ⁑ ( a k ) , a k , Ai ⁑ ( a k ) , k = 1 ⁒ ( 1 ) ⁒ 50 ; b k , Bi ⁑ ( b k ) , b k , Bi ⁑ ( b k ) , k = 1 ⁒ ( 1 ) ⁒ 20 . Precision is 8D. Entries for k = 1 ⁒ ( 1 ) ⁒ 20 are reproduced in Abramowitz and Stegun (1964, Chapter 10).

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  • Zhang and Jin (1996, p. 339) tabulates a k , Ai ⁑ ( a k ) , a k , Ai ⁑ ( a k ) , b k , Bi ⁑ ( b k ) , b k , Bi ⁑ ( b k ) , k = 1 ⁒ ( 1 ) ⁒ 20 ; 8D.

  • 8: Staff
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  • Ronald F. Boisvert, Editor at Large, NIST

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

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

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

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

  • 9: 3.4 Differentiation
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    B 2 5 = 1 120 ⁒ ( 6 10 ⁒ t 15 ⁒ t 2 + 20 ⁒ t 3 5 ⁒ t 4 ) ,
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    B 2 6 = 1 60 ⁒ ( 9 9 ⁒ t 30 ⁒ t 2 + 20 ⁒ t 3 + 5 ⁒ t 4 3 ⁒ t 5 ) ,
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    B 2 6 = 1 60 ⁒ ( 9 + 9 ⁒ t 30 ⁒ t 2 20 ⁒ t 3 + 5 ⁒ t 4 + 3 ⁒ t 5 ) ,
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    B 3 6 = 1 720 ⁒ ( 12 + 8 ⁒ t 45 ⁒ t 2 20 ⁒ t 3 + 15 ⁒ t 4 + 6 ⁒ t 5 ) .
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    B 3 7 = 1 720 ⁒ ( 48 + 8 ⁒ t 192 ⁒ t 2 20 ⁒ t 3 + 85 ⁒ t 4 + 6 ⁒ t 5 7 ⁒ t 6 ) ,
    10: Bibliography D
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  • B. Döring (1966) Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.