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1: 1.4 Calculus of One Variable
Chain Rule
L’Hôpital’s Rule
2: Bibliography
  • V. È. Adler (1994) Nonlinear chains and Painlevé equations. Phys. D 73 (4), pp. 335–351.
  • G. Allasia and R. Besenghi (1987a) Numerical computation of Tricomi’s psi function by the trapezoidal rule. Computing 39 (3), pp. 271–279.
  • G. Allasia and R. Besenghi (1991) Numerical evaluation of the Kummer function with complex argument by the trapezoidal rule. Rend. Sem. Mat. Univ. Politec. Torino 49 (3), pp. 315–327.
  • G. E. Andrews (2000) Umbral calculus, Bailey chains, and pentagonal number theorems. J. Combin. Theory Ser. A 91 (1-2), pp. 464–475.
  • G. E. Andrews (2001) Bailey’s Transform, Lemma, Chains and Tree. In Special Functions 2000: Current Perspective and Future Directions (Tempe, AZ), J. Bustoz, M. E. H. Ismail, and S. K. Suslov (Eds.), NATO Sci. Ser. II Math. Phys. Chem., Vol. 30, pp. 1–22.
  • 3: 1.5 Calculus of Two or More Variables
    Chain Rule
    4: 17.12 Bailey Pairs
    When (17.12.5) is iterated the resulting infinite sequence of Bailey pairs is called a Bailey Chain. … The Bailey pair and Bailey chain concepts have been extended considerably. …
    5: 10.73 Physical Applications
    §10.73(i) Bessel and Modified Bessel Functions
    Bessel functions first appear in the investigation of a physical problem in Daniel Bernoulli’s analysis of the small oscillations of a uniform heavy flexible chain. …
    6: 3.5 Quadrature
    §3.5(i) Trapezoidal Rules
    The composite trapezoidal rule is …
    §3.5(ii) Simpson’s Rule
    §3.5(iv) Interpolatory Quadrature Rules
    7: 3.8 Nonlinear Equations
    §3.8(ii) Newton’s Rule
    Newton’s rule is given by … Another iterative method is Halley’s rule: …The rule converges locally and is cubically convergent. …
    8: 8.25 Methods of Computation
    See Allasia and Besenghi (1987b) for the numerical computation of Γ ( a , z ) from (8.6.4) by means of the trapezoidal rule. … A numerical inversion procedure is also given for calculating the value of x (with 10S accuracy), when a and P ( a , x ) are specified, based on Newton’s rule3.8(ii)). …
    9: 9.17 Methods of Computation
    The trapezoidal rule3.5(i)) is then applied. … Zeros of the Airy functions, and their derivatives, can be computed to high precision via Newton’s rule3.8(ii)) or Halley’s rule3.8(v)), using values supplied by the asymptotic expansions of §9.9(iv) as initial approximations. …
    10: 7.22 Methods of Computation
    Additional references are Matta and Reichel (1971) for the application of the trapezoidal rule, for example, to the first of (7.7.2), and Gautschi (1970) and Cuyt et al. (2008) for continued fractions. …