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1: 12.10 Uniform Asymptotic Expansions for Large Parameter
§12.10 Uniform Asymptotic Expansions for Large Parameter
β–ΊLastly, the function g ⁑ ( ΞΌ ) in (12.10.3) and (12.10.4) has the asymptotic expansion: … β–ΊThe proof of the double asymptotic property then follows with the aid of error bounds; compare §10.41(iv). … β–Ί
2: 2.11 Remainder Terms; Stokes Phenomenon
β–Ί
§2.11(i) Numerical Use of Asymptotic Expansions
β–Ίβ–Ί
§2.11(ii) Connection Formulas
β–Ί
§2.11(iii) Exponentially-Improved Expansions
β–Ί
§2.11(vi) Direct Numerical Transformations
3: Bibliography F
β–Ί
  • FDLIBM (free C library)
  • β–Ί
  • S. Fempl (1960) Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
  • β–Ί
  • H. E. Fettis and J. C. Caslin (1964) Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.
  • β–Ί
  • C. L. Frenzen (1987b) On the asymptotic expansion of Mellin transforms. SIAM J. Math. Anal. 18 (1), pp. 273–282.
  • β–Ί
  • G. Freud (1969) On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.
  • 4: Bibliography V
    β–Ί
  • P. Verbeeck (1970) Rational approximations for exponential integrals E n ⁒ ( x ) . Acad. Roy. Belg. Bull. Cl. Sci. (5) 56, pp. 1064–1072.
  • β–Ί
  • R. VidΕ«nas and N. M. Temme (2002) Symbolic evaluation of coefficients in Airy-type asymptotic expansions. J. Math. Anal. Appl. 269 (1), pp. 317–331.
  • β–Ί
  • H. Volkmer and J. J. Wood (2014) A note on the asymptotic expansion of generalized hypergeometric functions. Anal. Appl. (Singap.) 12 (1), pp. 107–115.
  • β–Ί
  • H. Volkmer (2004a) Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.
  • β–Ί
  • H. Volkmer (2008) Approximation of eigenvalues of some differential equations by zeros of orthogonal polynomials. J. Comput. Appl. Math. 213 (2), pp. 488–500.
  • 5: Bibliography M
    β–Ί
  • A. J. MacLeod (1996b) Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.
  • β–Ί
  • Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.
  • β–Ί
  • J. W. Miles (1975) Asymptotic approximations for prolate spheroidal wave functions. Studies in Appl. Math. 54 (4), pp. 315–349.
  • β–Ί
  • D. S. Moak (1981) The q -analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.
  • β–Ί
  • C. Mortici (2013a) A continued fraction approximation of the gamma function. J. Math. Anal. Appl. 402 (2), pp. 405–410.
  • 6: Bibliography C
    β–Ί
  • B. C. Carlson and J. L. Gustafson (1994) Asymptotic approximations for symmetric elliptic integrals. SIAM J. Math. Anal. 25 (2), pp. 288–303.
  • β–Ί
  • R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.
  • β–Ί
  • E. W. Cheney (1982) Introduction to Approximation Theory. 2nd edition, Chelsea Publishing Co., New York.
  • β–Ί
  • W. J. Cody (1968) Chebyshev approximations for the Fresnel integrals. Math. Comp. 22 (102), pp. 450–453.
  • β–Ί
  • W. J. Cody (1969) Rational Chebyshev approximations for the error function. Math. Comp. 23 (107), pp. 631–637.
  • 7: 11.6 Asymptotic Expansions
    §11.6 Asymptotic Expansions
    β–Ί
    §11.6(i) Large | z | , Fixed Ξ½
    β–ΊMore fully, the series (11.2.1) and (11.2.2) can be regarded as generalized asymptotic expansions (§2.1(v)). … β–Ίand for an estimate of the relative error in this approximation see Watson (1944, p. 336).
    8: 28.8 Asymptotic Expansions for Large q
    §28.8 Asymptotic Expansions for Large q
    β–Ί
    Barrett’s Expansions
    β–Ί
    Dunster’s Approximations
    β–ΊDunster (1994a) supplies uniform asymptotic approximations for numerically satisfactory pairs of solutions of Mathieu’s equation (28.2.1). … β–Ί
    9: Bibliography B
    β–Ί
  • G. Backenstoss (1970) Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.
  • β–Ί
  • P. Baratella and L. Gatteschi (1988) The Bounds for the Error Term of an Asymptotic Approximation of Jacobi Polynomials. In Orthogonal Polynomials and Their Applications (Segovia, 1986), Lecture Notes in Math., Vol. 1329, pp. 203–221.
  • β–Ί
  • K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.
  • β–Ί
  • W. G. Bickley (1935) Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.
  • β–Ί
  • W. G. C. Boyd (1995) Approximations for the late coefficients in asymptotic expansions arising in the method of steepest descents. Methods Appl. Anal. 2 (4), pp. 475–489.
  • 10: 30.9 Asymptotic Approximations and Expansions
    §30.9 Asymptotic Approximations and Expansions
    β–ΊFor uniform asymptotic expansions in terms of Airy or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1986). … β–ΊFor uniform asymptotic expansions in terms of elementary, Airy, or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1992, 1995). … β–Ί
    §30.9(iii) Other Approximations and Expansions
    β–Ί