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1: 15.12 Asymptotic Approximations
§15.12(ii) Large c
For large b and c with c > b + 1 see López and Pagola (2011). … As λ , … If | ph z | < π , then as λ with | ph λ | π - δ , … For other extensions, see Wagner (1986), Temme (2003) and Temme (2015, Chapters 12 and 28).
2: Bibliography F
  • C. Ferreira, J. L. López, and E. Pérez Sinusía (2013a) The third Appell function for one large variable. J. Approx. Theory 165, pp. 60–69.
  • C. Ferreira, J. L. López, and E. Pérez Sinusía (2005) Incomplete gamma functions for large values of their variables. Adv. in Appl. Math. 34 (3), pp. 467–485.
  • C. Ferreira, J. L. López, and E. P. Sinusía (2013b) The second Appell function for one large variable. Mediterr. J. Math. 10 (4), pp. 1853–1865.
  • 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.
  • 3: 28.26 Asymptotic Approximations for Large q
    §28.26 Asymptotic Approximations for Large q
    §28.26(ii) Uniform Approximations
    4: 28.25 Asymptotic Expansions for Large z
    §28.25 Asymptotic Expansions for Large z
    5: Bibliography H
  • C. J. Howls and A. B. Olde Daalhuis (1999) On the resurgence properties of the uniform asymptotic expansion of Bessel functions of large order. Proc. Roy. Soc. London Ser. A 455, pp. 3917–3930.
  • 6: 28.34 Methods of Computation
  • (c)

    Use of asymptotic expansions for large z or large q . See §§28.25 and 28.26.

  • 7: 6.18 Methods of Computation
    A 0 , B 0 , and C 0 can be computed by Miller’s algorithm (§3.6(iii)), starting with initial values ( A N , B N , C N ) = ( 1 , 0 , 0 ) , say, where N is an arbitrary large integer, and normalizing via C 0 = 1 / z . …
    8: Bibliography B
  • L. Baker (1992) C Mathematical Function Handbook. McGraw-Hill, Inc., New York.
  • C. M. Bender and T. T. Wu (1973) Anharmonic oscillator. II. A study of perturbation theory in large order. Phys. Rev. D 7, pp. 1620–1636.
  • L. C. Biedenharn, R. L. Gluckstern, M. H. Hull, and G. Breit (1955) Coulomb functions for large charges and small velocities. Phys. Rev. (2) 97 (2), pp. 542–554.
  • 9: Bibliography J
  • S. Jorna and C. Springer (1971) Derivation of Green-type, transitional and uniform asymptotic expansions from differential equations. V. Angular oblate spheroidal wavefunctions p s ¯ n r ( η , h ) and q s ¯ n r ( η , h ) for large h . Proc. Roy. Soc. London Ser. A 321, pp. 545–555.
  • 10: 15.19 Methods of Computation
    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. …