About the Project

values%20at%20infinity

AdvancedHelp

(0.003 seconds)

1—10 of 12 matching pages

1: 6.16 Mathematical Applications
Hence, if x is fixed and n , then S n ( x ) 1 4 π , 0 , or 1 4 π according as 0 < x < π , x = 0 , or π < x < 0 ; compare (6.2.14). … The first maximum of 1 2 Si ( x ) for positive x occurs at x = π and equals ( 1.1789 ) × 1 4 π ; compare Figure 6.3.2. Hence if x = π / ( 2 n ) and n , then the limiting value of S n ( x ) overshoots 1 4 π by approximately 18%. Similarly if x = π / n , then the limiting value of S n ( x ) undershoots 1 4 π by approximately 10%, and so on. …
See accompanying text
Figure 6.16.2: The logarithmic integral li ( x ) , together with vertical bars indicating the value of π ( x ) for x = 10 , 20 , , 1000 . Magnify
2: 19.36 Methods of Computation
Complex values of the variables are allowed, with some restrictions in the case of R J that are sufficient but not always necessary. … Accurate values of F ( ϕ , k ) E ( ϕ , k ) for k 2 near 0 can be obtained from R D by (19.2.6) and (19.25.13). … This method loses significant figures in ρ if α 2 and k 2 are nearly equal unless they are given exact values—as they can be for tables. … The cases k c 2 / 2 p < and < p < k c 2 / 2 require different treatment for numerical purposes, and again precautions are needed to avoid cancellations. … 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). …
3: 10.75 Tables
  • 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).

  • Makinouchi (1966) tabulates all values of j ν , m and y ν , m in the interval ( 0 , 100 ) , with at least 29S. These are for ν = 0 ( 1 ) 5 , 10, 20; ν = 3 2 , 5 2 ; ν = m / n with m = 1 ( 1 ) n 1 and n = 3 ( 1 ) 8 , except for ν = 1 2 .

  • Zhang and Jin (1996, p. 270) tabulates 0 x J 0 ( t ) d t , 0 x t 1 ( 1 J 0 ( t ) ) d t , 0 x Y 0 ( t ) d t , x t 1 Y 0 ( t ) d t , x = 0 ( .1 ) 1 ( .5 ) 20 , 8D.

  • Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of K n ( z ) and K n ( z ) , for n = 2 ( 1 ) 20 , 9S.

  • 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 .

  • 4: 18.40 Methods of Computation
    There are many ways to implement these first two steps, noting that the expressions for α n and β n of equation (18.2.30) are of little practical numerical value, see Gautschi (2004) and Golub and Meurant (2010). … Results of low ( 2 to 3 decimal digits) precision for w ( x ) are easily obtained for N 10 to 20 . … The bottom and top of the steps at the x i are lower and upper bounds to a x i d μ ( x ) as made explicit via the Chebyshev inequalities discussed by Shohat and Tamarkin (1970, pp. 42–43). … This is a challenging case as the desired w RCP ( x ) on [ 1 , 1 ] has an essential singularity at x = 1 . … Achieving precisions at this level shown above requires higher than normal computational precision, see Gautschi (2009). …
    5: 5.11 Asymptotic Expansions
    Wrench (1968) gives exact values of g k up to g 20 . Spira (1971) corrects errors in Wrench’s results and also supplies exact and 45D values of g k for k = 21 , 22 , , 30 . … Lastly, as y ± , …uniformly for bounded real values of x . … If the sums in the expansions (5.11.1) and (5.11.2) are terminated at k = n 1 ( k 0 ) and z is real and positive, then the remainder terms are bounded in magnitude by the first neglected terms and have the same sign. …
    6: 26.10 Integer Partitions: Other Restrictions
    p m ( 𝒟 , n ) denotes the number of partitions of n into at most m distinct parts. p ( 𝒟 k , n ) denotes the number of partitions of n into parts with difference at least k . p ( 𝒟 3 , n ) denotes the number of partitions of n into parts with difference at least 3, except that multiples of 3 must differ by at least 6. … where the sum is over nonnegative integer values of k for which n 1 2 ( 3 k 2 ± k ) 0 . … The quantity A k ( n ) is real-valued. …
    7: 2.11 Remainder Terms; Stokes Phenomenon
    If the results agree within S significant figures, then it is likely—but not certain—that the truncated asymptotic series will yield at least S correct significant figures for larger values of x . … In this way we arrive at hyperasymptotic expansions. … 17408, compared with the correct valueComparison with the true valueFor example, using double precision d 20 is found to agree with (2.11.31) to 13D. …
    8: 12.11 Zeros
    If a > 1 2 , then V ( a , x ) has no positive real zeros, and if a = 3 2 2 n , n , then V ( a , x ) has a zero at x = 0 . … When a > 1 2 , U ( a , z ) has a string of complex zeros that approaches the ray ph z = 3 4 π as z , and a conjugate string. … For large negative values of a the real zeros of U ( a , x ) , U ( a , x ) , V ( a , x ) , and V ( a , x ) can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). …as μ ( = 2 a ) , s fixed. …
    12.11.9 u a , 1 2 1 2 μ ( 1 1.85575 708 μ 4 / 3 0.34438 34 μ 8 / 3 0.16871 5 μ 4 0.11414 μ 16 / 3 0.0808 μ 20 / 3 ) ,
    9: 25.12 Polylogarithms
    Other notations and names for Li 2 ( z ) include S 2 ( z ) (Kölbig et al. (1970)), Spence function Sp ( z ) (’t Hooft and Veltman (1979)), and L 2 ( z ) (Maximon (2003)). In the complex plane Li 2 ( z ) has a branch point at z = 1 . The principal branch has a cut along the interval [ 1 , ) and agrees with (25.12.1) when | z | 1 ; see also §4.2(i). …
    See accompanying text
    Figure 25.12.2: Absolute value of the dilogarithm function | Li 2 ( x + i y ) | , 20 x 20 , 20 y 20 . … Magnify 3D Help
    For other values of z , Li s ( z ) is defined by analytic continuation. …
    10: 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 (1935) Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.
  • G. Blanch and I. Rhodes (1955) Table of characteristic values of Mathieu’s equation for large values of the parameter. J. Washington Acad. Sci. 45 (6), pp. 166–196.
  • W. Bühring (1992) Generalized hypergeometric functions at unit argument. Proc. Amer. Math. Soc. 114 (1), pp. 145–153.