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1: Bibliography U
  • J. Urbanowicz (1988) On the equation f ( 1 ) 1 k + f ( 2 ) 2 k + + f ( x ) x k + R ( x ) = B y 2 . Acta Arith. 51 (4), pp. 349–368.
  • F. Ursell (1972) Integrals with a large parameter. Several nearly coincident saddle-points. Proc. Cambridge Philos. Soc. 72, pp. 49–65.
  • F. Ursell (1980) Integrals with a large parameter: A double complex integral with four nearly coincident saddle-points. Math. Proc. Cambridge Philos. Soc. 87 (2), pp. 249–273.
  • F. Ursell (1984) Integrals with a large parameter: Legendre functions of large degree and fixed order. Math. Proc. Cambridge Philos. Soc. 95 (2), pp. 367–380.
  • 2: Karl Dilcher
    Karl Dilcher (b. … Over the years he authored or coauthored numerous papers on Bernoulli numbers and related topics, and he maintains a large on-line bibliography on the subject. …
    3: 15.12 Asymptotic Approximations
    For large b and c with c > b + 1 see López and Pagola (2011). … If | ph ( z - 1 ) | < π , then as λ with | ph λ | π - δ , … If | ph z | < π , then as λ with | ph λ | π - δ , … If | ph z | < π , then as λ with | ph λ | 1 2 π - δ , … For other extensions, see Wagner (1986), Temme (2003) and Temme (2015, Chapters 12 and 28).
    4: 13.9 Zeros
    where n is a large positive integer, and the logarithm takes its principal value (§4.2(i)). … For fixed b and z in the large a -zeros of M ( a , b , z ) are given by …where n is a large positive integer. … For fixed b and z in the large a -zeros of U ( a , b , z ) are given by …where n is a large positive integer. …
    5: 8.11 Asymptotic Approximations and Expansions
    §8.11(i) Large z , Fixed a
    §8.11(ii) Large a , Fixed z
    §8.11(iii) Large a , Fixed z / a
    where …
    §8.11(iv) Large a , Bounded ( x - a ) / ( 2 a ) 1 2
    6: 13.8 Asymptotic Approximations for Large Parameters
    §13.8(i) Large | b | , Fixed a and z
    When the foregoing results are combined with Kummer’s transformation (13.2.39), an approximation is obtained for the case when | b | is large, and | b - a | and | z | are bounded.
    §13.8(ii) Large b and z , Fixed a and b / z
    §13.8(iii) Large a
    7: 15.19 Methods of Computation
    Large values of | a | or | b | , for example, delay convergence of the Gauss series, and may also lead to severe cancellation. For fast computation of F ( a , b ; c ; z ) with a , b and c complex, and with application to Pöschl–Teller–Ginocchio potential wave functions, see Michel and Stoitsov (2008). … The relations in §15.5(ii) can be used to compute F ( a , b ; c ; z ) , provided that care is taken to apply these relations in a stable manner; see §3.6(ii). Initial values for moderate values of | a | and | b | can be obtained by the methods of §15.19(i), and for large values of | a | , | b | , or | c | via the asymptotic expansions of §§15.12(ii) and 15.12(iii). For example, in the half-plane z 1 2 we can use (15.12.2) or (15.12.3) to compute F ( a , b ; c + N + 1 ; z ) and F ( a , b ; c + N ; z ) , where N is a large positive integer, and then apply (15.5.18) in the backward direction. …
    8: 16.22 Asymptotic Expansions
    Asymptotic expansions of G p , q m , n ( z ; a ; b ) for large z are given in Luke (1969a, §§5.7 and 5.10) and Luke (1975, §5.9). …
    9: Preface
    Bickel, B. …Eberhardt, B. … B. … B. …Undoubtedly, the editors have overlooked some individuals who contributed, as is inevitable in a large long-lasting project. …
    10: Bibliography F
  • B. R. Fabijonas, D. W. Lozier, and J. M. Rappoport (2003) Algorithms and codes for the Macdonald function: Recent progress and comparisons. J. Comput. Appl. Math. 161 (1), pp. 179–192.
  • S. Farid Khwaja and A. B. Olde Daalhuis (2014) Uniform asymptotic expansions for hypergeometric functions with large parameters IV. Anal. Appl. (Singap.) 12 (6), pp. 667–710.
  • H. E. Fettis (1970) On the reciprocal modulus relation for elliptic integrals. SIAM J. Math. Anal. 1 (4), pp. 524–526.
  • J. L. Fields (1973) Uniform asymptotic expansions of certain classes of Meijer G -functions for a large parameter. SIAM J. Math. Anal. 4 (3), pp. 482–507.
  • J. L. Fields (1983) Uniform asymptotic expansions of a class of Meijer G -functions for a large parameter. SIAM J. Math. Anal. 14 (6), pp. 1204–1253.