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1: 10.75 Tables
  • British Association for the Advancement of Science (1937) tabulates I 0 ( x ) , I 1 ( x ) , x = 0 ( .001 ) 5 , 7–8D; K 0 ( x ) , K 1 ( x ) , x = 0.01 ( .01 ) 5 , 7–10D; e x I 0 ( x ) , e x I 1 ( x ) , e x K 0 ( x ) , e x K 1 ( x ) , x = 5 ( .01 ) 10 ( .1 ) 20 , 8D. Also included are auxiliary functions to facilitate interpolation of the tables of K 0 ( x ) , K 1 ( x ) for small values of x .

  • 2: Bibliography B
  • G. Backenstoss (1970) Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.
  • K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.
  • W. G. Bickley and J. Nayler (1935) A short table of the functions Ki n ( x ) , from n = 1 to n = 16 . Phil. Mag. Series 7 20, pp. 343–347.
  • W. G. Bickley (1935) Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.
  • L. C. Biedenharn, R. L. Gluckstern, M. H. Hull, and G. Breit (1955) Coulomb functions for large charges and small velocities. Phys. Rev. (2) 97 (2), pp. 542–554.
  • 3: 9.7 Asymptotic Expansions
    Here δ denotes an arbitrary small positive constant and …Lastly, for x > 0 we define …For large x , … Numerical values of χ ( n ) are given in Table 9.7.1 for n = 1 ( 1 ) 20 to 2D. … As special cases, when 0 < x <
    4: 19.36 Methods of Computation
    When the differences are moderately small, the iteration is stopped, the elementary symmetric functions of certain differences are calculated, and a polynomial consisting of a fixed number of terms of the sum in (19.19.7) is evaluated. … Remedies for cancellation when x is real and near 0 are supplied in Midy (1975). … For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). … When the values of complete integrals are known, addition theorems with ψ = π / 2 19.11(ii)) ease the computation of functions such as F ( ϕ , k ) when 1 2 π ϕ is small and positive. Similarly, §19.26(ii) eases the computation of functions such as R F ( x , y , z ) when x ( > 0 ) is small compared with min ( y , z ) . …
    5: 28.35 Tables
  • 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 = 𝑏𝑜 n 2 q .

  • 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.

  • 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.

  • 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.

  • Ince (1932) includes the first zero for ce n , se n for n = 2 ( 1 ) 5 or 6 , q = 0 ( 1 ) 10 ( 2 ) 40 ; 4D. This reference also gives zeros of the first derivatives, together with expansions for small q .

  • 6: 26.6 Other Lattice Path Numbers
    M ( n ) is the number of lattice paths from ( 0 , 0 ) to ( n , n ) that stay on or above the line y = x and are composed of directed line segments of the form ( 2 , 0 ) , ( 0 , 2 ) , or ( 1 , 1 ) . … N ( n , k ) is the number of lattice paths from ( 0 , 0 ) to ( n , n ) that stay on or above the line y = x , are composed of directed line segments of the form ( 1 , 0 ) or ( 0 , 1 ) , and for which there are exactly k occurrences at which a segment of the form ( 0 , 1 ) is followed by a segment of the form ( 1 , 0 ) . … r ( n ) is the number of paths from ( 0 , 0 ) to ( n , n ) that stay on or above the diagonal y = x and are composed of directed line segments of the form ( 1 , 0 ) , ( 0 , 1 ) , or ( 1 , 1 ) . … For sufficiently small | x | and | y | , …
    26.6.8 n , k = 1 N ( n , k ) x n y k = 1 x x y ( 1 x x y ) 2 4 x 2 y 2 x ,
    7: 36.5 Stokes Sets
    Stokes sets are surfaces (codimension one) in 𝐱 space, across which Ψ K ( 𝐱 ; k ) or Ψ ( U ) ( 𝐱 ; k ) acquires an exponentially-small asymptotic contribution (in k ), associated with a complex critical point of Φ K or Φ ( U ) . … The Stokes set consists of the rays ph x = ± 2 π / 3 in the complex x -plane. … where x ± are the two smallest positive roots of the equation … When | X | > X 2 the Stokes set Y S ( X ) is given by … Alternatively, when | X | < X 2
    8: 10.73 Physical Applications
    Bessel functions first appear in the investigation of a physical problem in Daniel Bernoulli’s analysis of the small oscillations of a uniform heavy flexible chain. … See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25). … On separation of variables into cylindrical coordinates, the Bessel functions J n ( x ) , and modified Bessel functions I n ( x ) and K n ( x ) , all appear. … The analysis of the current distribution in circular conductors leads to the Kelvin functions ber x , bei x , ker x , and kei x . See Relton (1965, Chapter X, §§10.2, 10.3), Bowman (1958, Chapter III, §§51–53), McLachlan (1961, Chapters VIII and IX), and Russell (1909). …
    9: 11.6 Asymptotic Expansions
    where δ is an arbitrary small positive constant. …
    11.6.3 0 z 𝐊 0 ( t ) d t 2 π ( ln ( 2 z ) + γ ) 2 π k = 1 ( 1 ) k + 1 ( 2 k ) ! ( 2 k 1 ) ! ( k ! ) 2 ( 2 z ) 2 k , | ph z | π δ ,
    11.6.4 0 z 𝐌 0 ( t ) d t + 2 π ( ln ( 2 z ) + γ ) 2 π k = 1 ( 2 k ) ! ( 2 k 1 ) ! ( k ! ) 2 ( 2 z ) 2 k , | ph z | 1 2 π δ ,
    c 3 ( λ ) = 20 λ 6 4 λ 4 ,
    10: Bibliography L
  • A. Laforgia and M. E. Muldoon (1983) Inequalities and approximations for zeros of Bessel functions of small order. SIAM J. Math. Anal. 14 (2), pp. 383–388.
  • L. J. Landau (1999) Ratios of Bessel functions and roots of α J ν ( x ) + x J ν ( x ) = 0 . J. Math. Anal. Appl. 240 (1), pp. 174–204.
  • T. M. Larsen, D. Erricolo, and P. L. E. Uslenghi (2009) New method to obtain small parameter power series expansions of Mathieu radial and angular functions. Math. Comp. 78 (265), pp. 255–274.
  • P. W. Lawrence, R. M. Corless, and D. J. Jeffrey (2012) Algorithm 917: complex double-precision evaluation of the Wright ω function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.
  • D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.