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1: 10.75 Tables
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  • Olver (1960) tabulates j n , m , J n ⁑ ( j n , m ) , j n , m , J n ⁑ ( j n , m ) , y n , m , Y n ⁑ ( y n , m ) , y n , m , Y n ⁑ ( y n , m ) , n = 0 ⁒ ( 1 2 ) ⁒ 20 ⁀ 1 2 , m = 1 ⁒ ( 1 ) ⁒ 50 , 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as n ; see §10.21(viii), and more fully Olver (1954).

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  • Zhang and Jin (1996, pp. 296–305) tabulates 𝗃 n ⁑ ( x ) , 𝗃 n ⁑ ( x ) , 𝗒 n ⁑ ( x ) , 𝗒 n ⁑ ( x ) , 𝗂 n ( 1 ) ⁑ ( x ) , 𝗂 n ( 1 ) ⁑ ( x ) , 𝗄 n ⁑ ( x ) , 𝗄 n ⁑ ( x ) , n = 0 ⁒ ( 1 ) ⁒ 10 ⁒ ( 10 ) ⁒ 30 , 50, 100, x = 1 , 5, 10, 25, 50, 100, 8S; x ⁒ 𝗃 n ⁑ ( x ) , ( x ⁒ 𝗃 n ⁑ ( x ) ) , x ⁒ 𝗒 n ⁑ ( x ) , ( x ⁒ 𝗒 n ⁑ ( x ) ) (Riccati–Bessel functions and their derivatives), n = 0 ⁒ ( 1 ) ⁒ 10 ⁒ ( 10 ) ⁒ 30 , 50, 100, x = 1 , 5, 10, 25, 50, 100, 8S; real and imaginary parts of 𝗃 n ⁑ ( z ) , 𝗃 n ⁑ ( z ) , 𝗒 n ⁑ ( z ) , 𝗒 n ⁑ ( z ) , 𝗂 n ( 1 ) ⁑ ( z ) , 𝗂 n ( 1 ) ⁑ ( z ) , 𝗄 n ⁑ ( z ) , 𝗄 n ⁑ ( z ) , n = 0 ⁒ ( 1 ) ⁒ 15 , 20(10)50, 100, z = 4 + 2 ⁒ i , 20 + 10 ⁒ i , 8S. (For the notation replace j , y , i , k by 𝗃 , 𝗒 , 𝗂 ( 1 ) , 𝗄 , respectively.)

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  • Olver (1960) tabulates a n , m , 𝗃 n ⁑ ( a n , m ) , b n , m , 𝗒 n ⁑ ( b n , m ) , n = 1 ⁒ ( 1 ) ⁒ 20 , m = 1 ⁒ ( 1 ) ⁒ 50 , 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as n .

  • 2: 7.23 Tables
    §7.23 Tables
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  • Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of erf ⁑ z , x [ 0 , 5 ] , y = 0.5 ⁒ ( .5 ) ⁒ 3 , 7D and 8D, respectively; the real and imaginary parts of x e ± i ⁒ t 2 ⁒ d t , ( 1 / Ο€ ) ⁒ e βˆ“ i ⁒ ( x 2 + ( Ο€ / 4 ) ) ⁒ x e ± i ⁒ t 2 ⁒ d t , x = 0 ⁒ ( .5 ) ⁒ 20 ⁒ ( 1 ) ⁒ 25 , 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

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    §7.23(iv) Zeros
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  • Fettis et al. (1973) gives the first 100 zeros of erf ⁑ z and w ⁑ ( z ) (the table on page 406 of this reference is for w ⁑ ( z ) , not for erfc ⁑ z ), 11S.

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  • Zhang and Jin (1996, p. 642) includes the first 10 zeros of erf ⁑ z , 9D; the first 25 distinct zeros of C ⁑ ( z ) and S ⁑ ( z ) , 8S.

  • 3: 9.18 Tables
    §9.18 Tables
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    §9.18(iv) Zeros
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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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  • Sherry (1959) tabulates a k , Ai ⁑ ( a k ) , a k , Ai ⁑ ( a k ) , k = 1 ⁒ ( 1 ) ⁒ 50 ; 20S.

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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.

  • 4: 13.30 Tables
    §13.30 Tables
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  • 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.

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  • Abramowitz and Stegun (1964, Chapter 13) tabulates M ⁑ ( a , b , x ) for a = 1 ⁒ ( .1 ) ⁒ 1 , b = 0.1 ⁒ ( .1 ) ⁒ 1 , and x = 0.1 ⁒ ( .1 ) ⁒ 1 ⁒ ( 1 ) ⁒ 10 , 8S. Also the smallest positive x -zero of M ⁑ ( a , b , x ) for a = 1 ⁒ ( .1 ) 0.1 and b = 0.1 ⁒ ( .1 ) ⁒ 1 , 7D.

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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 ) ).

  • β–ΊFor other tables prior to 1961 see Fletcher et al. (1962) and Lebedev and Fedorova (1960).
    5: Bibliography F
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  • S. Fempl (1960) Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
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  • H. E. Fettis and J. C. Caslin (1964) Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.
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  • H. E. Fettis and J. C. Caslin (1973) Table errata; Complex zeros of Fresnel integrals. Math. Comp. 27 (121), pp. 219.
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  • A. Fletcher (1948) Guide to tables of elliptic functions. Math. Tables and Other Aids to Computation 3 (24), pp. 229–281.
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  • L. Fox (1960) Tables of Weber Parabolic Cylinder Functions and Other Functions for Large Arguments. National Physical Laboratory Mathematical Tables, Vol. 4. Department of Scientific and Industrial Research, Her Majesty’s Stationery Office, London.
  • 6: 12.19 Tables
    §12.19 Tables
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  • Murzewski and Sowa (1972) includes D n ⁑ ( x ) ( = U ⁑ ( n 1 2 , x ) ) for n = 1 ⁒ ( 1 ) ⁒ 20 , x = 0 ⁒ ( .05 ) ⁒ 3 , 7S.

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  • Zhang and Jin (1996, pp. 455–473) includes U ⁑ ( ± n 1 2 , x ) , V ⁑ ( ± n 1 2 , x ) , U ⁑ ( ± Ξ½ 1 2 , x ) , V ⁑ ( ± Ξ½ 1 2 , x ) , and derivatives, Ξ½ = n + 1 2 , n = 0 ⁒ ( 1 ) ⁒ 10 ⁒ ( 10 ) ⁒ 30 , x = 0.5 , 1 , 5 , 10 , 30 , 50 , 8S; W ⁑ ( a , ± x ) , W ⁑ ( a , ± x ) , and derivatives, a = h ⁒ ( 1 ) ⁒ 5 + h , x = 0.5 , 1 and a = h ⁒ ( 1 ) ⁒ 5 + h , x = 5 , h = 0 , 0.5 , 8S. Also, first zeros of U ⁑ ( a , x ) , V ⁑ ( a , x ) , and of derivatives, a = 6 ⁒ ( .5 ) 1 , 6D; first three zeros of W ⁑ ( a , x ) and of derivative, a = 0 ⁒ ( .5 ) ⁒ 4 , 6D; first three zeros of W ⁑ ( a , ± x ) and of derivative, a = 0.5 ⁒ ( .5 ) ⁒ 5.5 , 6D; real and imaginary parts of U ⁑ ( a , z ) , a = 1.5 ⁒ ( 1 ) ⁒ 1.5 , z = x + i ⁒ y , x = 0.5 , 1 , 5 , 10 , y = 0 ⁒ ( .5 ) ⁒ 10 , 8S.

  • β–ΊFor other tables prior to 1961 see Fletcher et al. (1962) and Lebedev and Fedorova (1960).
    7: Bibliography S
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  • K. L. Sala (1989) Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.
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  • J. Segura (2001) Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros. Math. Comp. 70 (235), pp. 1205–1220.
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  • A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
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  • A. Stankiewicz (1968) Tables of the integro-exponential functions. Acta Astronom. 18, pp. 289–311.
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  • J. R. Stembridge (1995) A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.
  • 8: Bibliography L
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  • D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.
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  • D. J. Leeming (1989) The real zeros of the Bernoulli polynomials. J. Approx. Theory 58 (2), pp. 124–150.
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  • Y. L. Luke (1971b) Miniaturized tables of Bessel functions. Math. Comp. 25 (114), pp. 323–330.
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  • Y. L. Luke (1972) Miniaturized tables of Bessel functions. III. Math. Comp. 26 (117), pp. 237–240 and A14–B5.
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  • W. Luther (1995) Highly accurate tables for elementary functions. BIT 35 (3), pp. 352–360.
  • 9: 28.35 Tables
    §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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  • Kirkpatrick (1960) contains tables of the modified functions Ce n ⁑ ( x , q ) , Se n + 1 ⁑ ( x , q ) for n = 0 ⁒ ( 1 ) ⁒ 5 , q = 1 ⁒ ( 1 ) ⁒ 20 , x = 0.1 ⁒ ( .1 ) ⁒ 1 ; 4D or 5D.

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    §28.35(iii) Zeros
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    §28.35(iv) Further Tables
    10: Bibliography I
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  • Y. Ikebe, Y. Kikuchi, and I. Fujishiro (1991) Computing zeros and orders of Bessel functions. J. Comput. Appl. Math. 38 (1-3), pp. 169–184.
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  • Y. Ikebe (1975) The zeros of regular Coulomb wave functions and of their derivatives. Math. Comp. 29, pp. 878–887.
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  • E. L. Ince (1932) Tables of the elliptic cylinder functions. Proc. Roy. Soc. Edinburgh Sect. A 52, pp. 355–433.
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  • K. Inkeri (1959) The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.
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  • M. E. H. Ismail and M. E. Muldoon (1995) Bounds for the small real and purely imaginary zeros of Bessel and related functions. Methods Appl. Anal. 2 (1), pp. 1–21.