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11: Bibliography W
  • E. Wagner (1988) Asymptotische Entwicklungen der hypergeometrischen Funktion F ( a , b , c , z ) für | c | und konstante Werte a , b und z . Demonstratio Math. 21 (2), pp. 441–458 (German).
  • T. Watanabe, M. Natori, and T. Oguni (Eds.) (1994) Mathematical Software for the P.C. and Work Stations – A Collection of Fortran 77 Programs. North-Holland Publishing Co., Amsterdam.
  • M. I. Weinstein and J. B. Keller (1985) Hill’s equation with a large potential. SIAM J. Appl. Math. 45 (2), pp. 200–214.
  • J. A. Wheeler (1937) Wave functions for large arguments by the amplitude-phase method. Phys. Rev. 52, pp. 1123–1127.
  • R. Wong (1973a) An asymptotic expansion of W k , m ( z ) with large variable and parameters. Math. Comp. 27 (122), pp. 429–436.
  • 12: Bibliography Z
  • A. Zarzo, J. S. Dehesa, and R. J. Yañez (1995) Distribution of zeros of Gauss and Kummer hypergeometric functions. A semiclassical approach. Ann. Numer. Math. 2 (1-4), pp. 457–472.
  • S. Zhang and J. Jin (1996) Computation of Special Functions. John Wiley & Sons Inc., New York.
  • Ya. M. Zhileĭkin and A. B. Kukarkin (1995) A fast Fourier-Bessel transform algorithm. Zh. Vychisl. Mat. i Mat. Fiz. 35 (7), pp. 1128–1133 (Russian).
  • C. H. Ziener, M. Rückl, T. Kampf, W. R. Bauer, and H. P. Schlemmer (2012) Mathieu functions for purely imaginary parameters. J. Comput. Appl. Math. 236 (17), pp. 4513–4524.
  • D. G. Zill and B. C. Carlson (1970) Symmetric elliptic integrals of the third kind. Math. Comp. 24 (109), pp. 199–214.
  • 13: 10.70 Zeros
    If m is a large positive integer, then …
    14: 8.18 Asymptotic Expansions of I x ( a , b )
    §8.18 Asymptotic Expansions of I x ( a , b )
    If b and x are fixed, with b > 0 and 0 < x < 1 , then as a
    Large a , Fixed b
    Then as a + b , … For asymptotic expansions for large values of a and/or b of the x -solution of the equation …
    15: Bibliography N
  • M. Nardin, W. F. Perger, and A. Bhalla (1992a) Algorithm 707: CONHYP: A numerical evaluator of the confluent hypergeometric function for complex arguments of large magnitudes. ACM Trans. Math. Software 18 (3), pp. 345–349.
  • M. Nardin, W. F. Perger, and A. Bhalla (1992b) Numerical evaluation of the confluent hypergeometric function for complex arguments of large magnitudes. J. Comput. Appl. Math. 39 (2), pp. 193–200.
  • M. Newman (1967) Solving equations exactly. J. Res. Nat. Bur. Standards Sect. B 71B, pp. 171–179.
  • E. W. Ng and M. Geller (1969) A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
  • Numerical Recipes (commercial C, C++, Fortran 77, and Fortran 90 libraries)
  • 16: About the Project
    These products resulted from the leadership of the Editors and Associate Editors pictured in Figure 1; the contributions of 29 authors, 10 validators, and 5 principal developers; and assistance from a large group of contributing developers, consultants, assistants and interns. …
    17: 8.12 Uniform Asymptotic Expansions for Large Parameter
    8.12.18 Q ( a , z ) P ( a , z ) } z a - 1 2 e - z Γ ( a ) ( d ( ± χ ) k = 0 A k ( χ ) z k / 2 k = 1 B k ( χ ) z k / 2 ) ,
    B 1 ( χ ) = 1 3 + 1 6 χ 2 .
    Higher coefficients A k ( χ ) , B k ( χ ) , up to k = 8 , are given in Paris (2002b). Lastly, a uniform approximation for Γ ( a , a x ) for large a , with error bounds, can be found in Dunster (1996a). …
    18: 2.3 Integrals of a Real Variable
    converges for all sufficiently large x , and q ( t ) is infinitely differentiable in a neighborhood of the origin. … When p ( t ) is real and x is a large positive parameter, the main contribution to the integral … When the parameter x is large the contributions from the real and imaginary parts of the integrand in … If p ( b ) is finite, then both endpoints contribute: … k ( ) and λ are positive constants, α is a variable parameter in an interval α 1 α α 2 with α 1 0 and 0 < α 2 k , and x is a large positive parameter. …
    19: Bibliography L
  • 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.
  • H. T. Lau (1995) A Numerical Library in C for Scientists and Engineers. CRC Press, Boca Raton, FL.
  • H. T. Lau (2004) A Numerical Library in Java for Scientists & Engineers. Chapman & Hall/CRC, Boca Raton, FL.
  • 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.
  • J. L. López (1999) Asymptotic expansions of the Whittaker functions for large order parameter. Methods Appl. Anal. 6 (2), pp. 249–256.
  • 20: Bibliography D
  • B. Deconinck and M. van Hoeij (2001) Computing Riemann matrices of algebraic curves. Phys. D 152/153, pp. 28–46.
  • B. I. Dunlap and B. R. Judd (1975) Novel identities for simple n - j symbols. J. Mathematical Phys. 16, pp. 318–319.
  • 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.
  • T. M. Dunster (1991) Conical functions with one or both parameters large. Proc. Roy. Soc. Edinburgh Sect. A 119 (3-4), pp. 311–327.
  • 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.