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11: Bibliography N
  • NAG (commercial C and Fortran libraries) Numerical Algorithms Group, Ltd..
  • Numerical Recipes (commercial C, C++, Fortran 77, and Fortran 90 libraries)
  • 12: Bibliography U
  • F. Ursell (1960) On Kelvin’s ship-wave pattern. J. Fluid Mech. 8 (3), pp. 418–431.
  • 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.
  • 13: 10.21 Zeros
    §10.21(vi) McMahon’s Asymptotic Expansions for Large Zeros
    §10.21(vii) Asymptotic Expansions for Large Order
    §10.21(viii) Uniform Asymptotic Approximations for Large Order
    The asymptotic expansion of the large positive zeros (not necessarily the m th) of the function …
    14: Bibliography I
    15: 2.10 Sums and Sequences
    for large n . … As a first estimate for large n (5.11.7) shows that the integrals around the large quarter circles vanish as n . …
    Example
    In Condition (c) we have …
    16: Bibliography L
  • 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.
  • 17: 2.3 Integrals of a Real Variable
  • (c)

    The integral (2.3.13) converges absolutely for all sufficiently large x .

  • 18: 18.15 Asymptotic Approximations
    For large β , fixed α , and 0 n / β c , Dunster (1999) gives asymptotic expansions of P n ( α , β ) ( z ) that are uniform in unbounded complex z -domains containing z = ± 1 . …This reference also supplies asymptotic expansions of P n ( α , β ) ( z ) for large n , fixed α , and 0 β / n c . …
    19: 33.10 Limiting Forms for Large ρ or Large | η |
    §33.10 Limiting Forms for Large ρ or Large | η |
    §33.10(i) Large ρ
    §33.10(ii) Large Positive η
    §33.10(iii) Large Negative η
    C 0 ( η ) ( - 2 π η ) 1 / 2 .
    20: 33.5 Limiting Forms for Small ρ , Small | η | , or Large
    §33.5 Limiting Forms for Small ρ , Small | η | , or Large
    F ( η , ρ ) C ( η ) ρ + 1 ,
    F ( η , ρ ) ( + 1 ) C ( η ) ρ .
    33.5.6 C ( 0 ) = 2 ! ( 2 + 1 ) ! = 1 ( 2 + 1 ) !! .
    §33.5(iv) Large