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21: 22.19 Physical Applications
Its dynamics for purely imaginary time is connected to the theory of instantons (Itzykson and Zuber (1980, p. 572), Schäfer and Shuryak (1998)), to WKB theory, and to large-order perturbation theory (Bender and Wu (1973), Simon (1982)). …
22: Bibliography U
  • 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.
  • 23: Bibliography S
  • B. Simon (1982) Large orders and summability of eigenvalue perturbation theory: A mathematical overview. Int. J. Quantum Chem. 21, pp. 3–25.
  • 24: 14.20 Conical (or Mehler) Functions
    §14.20(ix) Asymptotic Approximations: Large μ , 0 τ A μ
    25: Bibliography B
  • 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.
  • 26: 10.40 Asymptotic Expansions for Large Argument
    ν -Derivative
    §10.40(ii) Error Bounds for Real Argument and Order
    §10.40(iii) Error Bounds for Complex Argument and Order
    27: 10.17 Asymptotic Expansions for Large Argument
    §10.17(iii) Error Bounds for Real Argument and Order
    28: 3.5 Quadrature
    If k in (3.5.4) is not arbitrarily large, and if odd-order derivatives of f are known at the end points a and b , then the composite trapezoidal rule can be improved by means of the Euler–Maclaurin formula (§2.10(i)). …
    29: 3.8 Nonlinear Equations
    for all n sufficiently large, where A and p are independent of n , then the sequence is said to have convergence of the p th order. …
    30: 12.11 Zeros
    §12.11(ii) Asymptotic Expansions of Large Zeros
    When a > - 1 2 the zeros are asymptotically given by z a , s and z a , s ¯ , where s is a large positive integer and …
    §12.11(iii) Asymptotic Expansions for Large Parameter
    For large negative values of a the real zeros of U ( a , x ) , U ( a , x ) , V ( a , x ) , and V ( a , x ) can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). For example, let the s th real zeros of U ( a , x ) and U ( a , x ) , counted in descending order away from the point z = 2 - a , be denoted by u a , s and u a , s , respectively. …