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large a (or b) and c

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1: Bibliography U
  • 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
    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
    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: 10.70 Zeros
    If m is a large positive integer, then …
    5: 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. …
    6: 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
    §8.11(iv) Large a , Bounded ( x - a ) / ( 2 a ) 1 2
    7: 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. …
    8: 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
    9: 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. … 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. …
    10: 10.72 Mathematical Applications
    where z is a real or complex variable and u is a large real or complex parameter. … In regions in which (10.72.1) has a simple turning point z 0 , that is, f ( z ) and g ( z ) are analytic (or with weaker conditions if z = x is a real variable) and z 0 is a simple zero of f ( z ) , asymptotic expansions of the solutions w for large u can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order 1 3 9.6(i)). … In regions in which the function f ( z ) has a simple pole at z = z 0 and ( z - z 0 ) 2 g ( z ) is analytic at z = z 0 (the case λ = - 1 in §10.72(i)), asymptotic expansions of the solutions w of (10.72.1) for large u can be constructed in terms of Bessel functions and modified Bessel functions of order ± 1 + 4 ρ , where ρ is the limiting value of ( z - z 0 ) 2 g ( z ) as z z 0 . …