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21: Bibliography L
β–Ί
  • R. E. Langer (1934) The solutions of the Mathieu equation with a complex variable and at least one parameter large. Trans. Amer. Math. Soc. 36 (3), pp. 637–695.
  • β–Ί
  • H. T. Lau (1995) A Numerical Library in C for Scientists and Engineers. CRC Press, Boca Raton, FL.
  • β–Ί
  • H. T. Lau (2004) A Numerical Library in Java for Scientists & Engineers. Chapman & Hall/CRC, Boca Raton, FL.
  • β–Ί
  • J. L. López and P. J. Pagola (2010) The confluent hypergeometric functions M ⁒ ( a , b ; z ) and U ⁒ ( a , b ; z ) for large b and z . J. Comput. Appl. Math. 233 (6), pp. 1570–1576.
  • β–Ί
  • J. L. López (1999) Asymptotic expansions of the Whittaker functions for large order parameter. Methods Appl. Anal. 6 (2), pp. 249–256.
  • 22: Bibliography D
    β–Ί
  • T. M. Dunster (1990a) Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter. SIAM J. Math. Anal. 21 (4), pp. 995–1018.
  • β–Ί
  • T. M. Dunster (1991) Conical functions with one or both parameters large. Proc. Roy. Soc. Edinburgh Sect. A 119 (3-4), pp. 311–327.
  • β–Ί
  • T. M. Dunster (2003b) Uniform asymptotic expansions for associated Legendre functions of large order. Proc. Roy. Soc. Edinburgh Sect. A 133 (4), pp. 807–827.
  • 23: 2.3 Integrals of a Real Variable
    β–Ίconverges for all sufficiently large x , and q ⁑ ( t ) is infinitely differentiable in a neighborhood of the origin. … β–ΊAlternatively, assume b = , q ⁑ ( t ) is infinitely differentiable on [ a , ) , and each of the integrals e i ⁒ x ⁒ t ⁒ q ( s ) ⁑ ( t ) ⁒ d t , s = 0 , 1 , 2 , , converges as t uniformly for all sufficiently large x . … β–ΊWhen p ⁑ ( t ) is real and x is a large positive parameter, the main contribution to the integral … β–ΊWhen the parameter x is large the contributions from the real and imaginary parts of the integrand in … β–Ί k ( ) and Ξ» are positive constants, Ξ± is a variable parameter in an interval Ξ± 1 Ξ± Ξ± 2 with Ξ± 1 0 and 0 < Ξ± 2 k , and x is a large positive parameter. …
    24: Bibliography T
    β–Ί
  • N. M. Temme (1994b) Computational aspects of incomplete gamma functions with large complex parameters. In Approximation and Computation. A Festschrift in Honor of Walter Gautschi, R. V. M. Zahar (Ed.), International Series of Numerical Mathematics, Vol. 119, pp. 551–562.
  • β–Ί
  • W. J. Thompson (1997) Atlas for Computing Mathematical Functions: An Illustrated Guide for Practitioners. John Wiley & Sons Inc., New York.
  • 25: 27.16 Cryptography
    β–ΊFor example, a code maker chooses two large primes p and q of about 400 decimal digits each. …
    26: 2.4 Contour Integrals
    β–ΊFor 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 … β–Ίin which z is a large real or complex parameter, p ⁑ ( Ξ± , t ) and q ⁑ ( Ξ± , t ) are analytic functions of t and continuous in t and a second parameter Ξ± . … β–ΊFor large | z | , I ⁑ ( Ξ± , z ) is approximated uniformly by the integral that corresponds to (2.4.19) when f ⁑ ( Ξ± , w ) is replaced by a constant. …
    27: 19.36 Methods of Computation
    β–ΊDescending Gauss transformations of Ξ  ⁑ ( Ο• , Ξ± 2 , k ) (see (19.8.20)) are used in Fettis (1965) to compute a large table (see §19.37(iii)). …
    28: Bibliography C
    β–Ί
  • F. Calogero (1978) Asymptotic behaviour of the zeros of the (generalized) Laguerre polynomial L n Ξ± ⁒ ( x )  as the index Ξ±  and limiting formula relating Laguerre polynomials of large index and large argument to Hermite polynomials. Lett. Nuovo Cimento (2) 23 (3), pp. 101–102.
  • β–Ί
  • 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.
  • 29: 2.11 Remainder Terms; Stokes Phenomenon
    β–Ίwith m a large integer. … β–ΊFor large ρ the integrand has a saddle point at t = e i ⁒ ΞΈ . …
    30: Bibliography G