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21: 2.4 Contour Integrals
Then … For large t , the asymptotic expansion of q ( t ) may be obtained from (2.4.3) by Haar’s method. This depends on the availability of a comparison function F ( z ) for Q ( z ) that has an inverse transform … Let 𝒫 denote the path for the contour integral … in which z is a large real or complex parameter, p ( α , t ) and q ( α , t ) are analytic functions of t and continuous in t and a second parameter α . … For large | z | , I ( α , z ) is approximated uniformly by the integral that corresponds to (2.4.19) when f ( α , w ) is replaced by a constant. …
22: 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. … Descending Gauss transformations of Π ( ϕ , α 2 , k ) (see (19.8.20)) are used in Fettis (1965) to compute a large table (see §19.37(iii)). … The function cel ( k c , p , a , b ) is computed by successive Bartky transformations (Bulirsch and Stoer (1968), Bulirsch (1969b)). The function el2 ( x , k c , a , b ) is computed by descending Landen transformations if x is real, or by descending Gauss transformations if x is complex (Bulirsch (1965b)). … Bulirsch (1969a, b) extend Bartky’s transformation to el3 ( x , k c , p ) by expressing it in terms of the first incomplete integral, a complete integral of the third kind, and a more complicated integral to which Bartky’s method can be applied. …
23: Bibliography C
  • F. Calogero (1978) Asymptotic behaviour of the zeros of the (generalized) Laguerre polynomial L n α ( x )  as the index α  and limiting formula relating Laguerre polynomials of large index and large argument to Hermite polynomials. Lett. Nuovo Cimento (2) 23 (3), pp. 101–102.
  • B. C. Carlson (1970) Inequalities for a symmetric elliptic integral. Proc. Amer. Math. Soc. 25 (3), pp. 698–703.
  • B. C. Carlson (1972b) Intégrandes à deux formes quadratiques. C. R. Acad. Sci. Paris Sér. AB 274 (15 May, 1972, Sér. A), pp. 1458–1461 (French).
  • J. Chen (1966) On the representation of a large even integer as the sum of a prime and the product of at most two primes. Kexue Tongbao (Foreign Lang. Ed.) 17, pp. 385–386.
  • A. Cuyt, V. Petersen, B. Verdonk, H. Waadeland, W. B. Jones, and C. Bonan-Hamada (2007) Handbook of Continued Fractions for Special Functions. Kluwer Academic Publishers Group, Dordrecht.
  • 24: Bibliography G
  • B. Gabutti and B. Minetti (1981) A new application of the discrete Laguerre polynomials in the numerical evaluation of the Hankel transform of a strongly decreasing even function. J. Comput. Phys. 42 (2), pp. 277–287.
  • B. Gabutti (1980) On the generalization of a method for computing Bessel function integrals. J. Comput. Appl. Math. 6 (2), pp. 167–168.
  • M. Geller and E. W. Ng (1969) A table of integrals of the exponential integral. J. Res. Nat. Bur. Standards Sect. B 73B, pp. 191–210.
  • M. Geller and E. W. Ng (1971) A table of integrals of the error function. II. Additions and corrections. J. Res. Nat. Bur. Standards Sect. B 75B, pp. 149–163.
  • GSL (free C library) GNU Scientific Library The GNU Project.
  • 25: 17.2 Calculus
    Note that (17.2.6_1) is just (27.14.14) with a = d = 0 and b = c = 1 . … When a = q m + 1 , where m is a nonnegative integer, (17.2.37) reduces to the q -binomial series … If f ( x ) is continuous on [ 0 , a ] , then … These identities are the first in a large collection of similar results. …
    26: 2.11 Remainder Terms; Stokes Phenomenon
    with m a large integer. … In order to guard against this kind of error remaining undetected, the wanted function may need to be computed by another method (preferably nonasymptotic) for the smallest value of the (large) asymptotic variable x that is intended to be used. … However, on combining (2.11.6) with the connection formula (8.19.18), with m = 1 , we derive … For large ρ the integrand has a saddle point at t = e i θ . … For large | z | , with | ph z | 3 2 π δ ( < 3 2 π ), the Whittaker function of the second kind has the asymptotic expansion (§13.19) …
    27: Bibliography O
  • A. B. Olde Daalhuis (2003a) Uniform asymptotic expansions for hypergeometric functions with large parameters. I. Analysis and Applications (Singapore) 1 (1), pp. 111–120.
  • A. B. Olde Daalhuis (2003b) Uniform asymptotic expansions for hypergeometric functions with large parameters. II. Analysis and Applications (Singapore) 1 (1), pp. 121–128.
  • A. B. Olde Daalhuis (2010) Uniform asymptotic expansions for hypergeometric functions with large parameters. III. Analysis and Applications (Singapore) 8 (2), pp. 199–210.
  • 28: 12.2 Differential Equations
    For real values of z ( = x ) , numerically satisfactory pairs of solutions (§2.7(iv)) of (12.2.2) are U ( a , x ) and V ( a , x ) when x is positive, or U ( a , x ) and V ( a , x ) when x is negative. For (12.2.3) W ( a , x ) and W ( a , x ) comprise a numerically satisfactory pair, for all x . … Its importance is that when a is negative and | a | is large, U ( a , x ) and U ¯ ( a , x ) asymptotically have the same envelope (modulus) and are 1 2 π out of phase in the oscillatory interval 2 a < x < 2 a . … In the oscillatory interval we define … where F ( a , x ) ( > 0), θ ( a , x ) , G ( a , x ) ( > 0), and ψ ( a , x ) are real. …
    29: 18.2 General Orthogonal Polynomials
    Let ( a , b ) be a finite or infinite open interval in . … Assume that the interval [ a , b ] is bounded. …
    The Nevai class 𝐌 ( a , b )
    For a large class of OP’s p n there exist pairs of differentiation formulas … For OP’s p n on [ a , b ] with weight function w ( x ) and orthogonality relation (18.2.5_5) assume that b < and w ( x ) is non-decreasing in the interval [ a , b ] . …
    30: Bibliography S
  • B. Simon (1982) Large orders and summability of eigenvalue perturbation theory: A mathematical overview. Int. J. Quantum Chem. 21, pp. 3–25.