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11: Bibliography W
  • H. S. Wall (1948) Analytic Theory of Continued Fractions. D. Van Nostrand Company, Inc., New York.
  • 12: 8.17 Incomplete Beta Functions
    However, in the case of §8.17 it is straightforward to continue most results analytically to other real values of a , b , and x , and also to complex values. …
    13: 1.12 Continued Fractions
    For analytical and numerical applications of continued fractions to special functions see §3.10. …
    14: 3.8 Nonlinear Equations
    This is an iterative method for real twice-continuously differentiable, or complex analytic, functions: …
    15: 33.23 Methods of Computation
    Thompson and Barnett (1985, 1986) and Thompson (2004) use combinations of series, continued fractions, and Padé-accelerated asymptotic expansions (§3.11(iv)) for the analytic continuations of Coulomb functions. …
    16: 4.37 Inverse Hyperbolic Functions
    Elsewhere on the integration paths in (4.37.1) and (4.37.2) the branches are determined by continuity. … These functions are analytic in the cut plane depicted in Figure 4.37.1(iv), (v), (vi), respectively. … It should be noted that the imaginary axis is not a cut; the function defined by (4.37.19) and (4.37.20) is analytic everywhere except on ( - , 1 ] . …
    17: 4.12 Generalized Logarithms and Exponentials
    Both ϕ ( x ) and ψ ( x ) are continuously differentiable. … For analytic generalized logarithms, see Kneser (1950).
    18: Frank W. J. Olver
    Olver continued to maintain a connection to NIST after moving to the university. …
  • He continued his editing work until the time of his death on April 22, 2013 at age 88.
    19: 10.72 Mathematical Applications
    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)). … The number m can also be replaced by any real constant λ ( > - 2 ) in the sense that ( z - z 0 ) - λ f ( z ) is analytic and nonvanishing at z 0 ; moreover, g ( z ) is permitted to have a single or double pole at z 0 . … 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 . … In (10.72.1) assume f ( z ) = f ( z , α ) and g ( z ) = g ( z , α ) depend continuously on a real parameter α , f ( z , α ) has a simple zero z = z 0 ( α ) and a double pole z = 0 , except for a critical value α = a , where z 0 ( a ) = 0 . Assume that whether or not α = a , z 2 g ( z , α ) is analytic at z = 0 . …
    20: 31.11 Expansions in Series of Hypergeometric Functions
    For example, consider the Heun function which is analytic at z = a and has exponent α at . …In this case the accessory parameter q is a root of the continued-fraction equation …