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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: 18.40 Methods of Computation
    The question is then: how is this possible given only F N ( z ) , rather than F ( z ) itself? F N ( z ) often converges to smooth results for z off the real axis for z at a distance greater than the pole spacing of the x n , this may then be followed by approximate numerical analytic continuation via fitting to lower order continued fractions (either Padé, see §3.11(iv), or pointwise continued fraction approximants, see Schlessinger (1968, Appendix)), to F N ( z ) and evaluating these on the real axis in regions of higher pole density that those of the approximating function. …
    16: 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. …
    17: 4.37 Inverse Hyperbolic Functions
    4.37.6 Arccoth z = Arctanh ( 1 / z ) .
    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 ] . …
    18: 4.12 Generalized Logarithms and Exponentials
    Both ϕ ( x ) and ψ ( x ) are continuously differentiable. … For analytic generalized logarithms, see Kneser (1950).
    19: 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.
    20: 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 . …