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11: 3.4 Differentiation
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►where is as in (3.3.10).
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►If can be extended analytically into the complex plane, then from Cauchy’s integral formula (§1.9(iii))
…Taking to be a circle of radius centered at , we obtain
…The integral on the right-hand side can be approximated by the composite trapezoidal rule (3.5.2).
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►As explained in §§3.5(i) and 3.5(ix) the composite trapezoidal rule can be very efficient for computing integrals with analytic periodic integrands.
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12: 10.74 Methods of Computation
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►Furthermore, the attainable accuracy can be increased substantially by use of the exponentially-improved expansions given in §10.17(v), even more so by application of the hyperasymptotic expansions to be found in the references in that subsection.
►For large positive real values of the uniform asymptotic expansions of §§10.20(i) and 10.20(ii) can be used.
…It should be noted, however, that there is a difficulty in evaluating the coefficients , , , and , from the explicit expressions (10.20.10)–(10.20.13) when is close to owing to severe cancellation.
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►And since there are no error terms they could, in theory, be used for all values of ; however, there may be severe cancellation when is not large compared with .
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►Then and can be generated by either forward or backward recurrence on when , but if then to maintain stability has to be generated by backward recurrence on , and has to be generated by forward recurrence on .
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13: 16.25 Methods of Computation
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►Instead a boundary-value problem needs to be formulated and solved.
See §§3.6(vii), 3.7(iii), Olde Daalhuis and Olver (1998), Lozier (1980), and Wimp (1984, Chapters 7, 8).
14: 36 Integrals with Coalescing Saddles
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15: 19.36 Methods of Computation
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►The cancellations can be eliminated, however, by using (19.25.10).
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can be evaluated by using (19.25.5).
can be evaluated by using (19.25.7), and by using (19.21.10), but cancellations may become significant.
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►If , then the method fails, but the function can be expressed by (19.6.13) in terms of , for which Neuman (1969b) uses ascending Landen transformations.
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►Faster convergence of power series for and can be achieved by using (19.5.1) and (19.5.2) in the right-hand sides of (19.8.12).
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16: Bibliography D
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Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters.
J. Number Theory 25 (1), pp. 72–80.
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Handbuch der Laplace-Transformation. Bd. II. Anwendungen der Laplace-Transformation. 1. Abteilung.
Birkhäuser Verlag, Basel und Stuttgart (German).
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Theta functions and non-linear equations.
Uspekhi Mat. Nauk 36 (2(218)), pp. 11–80 (Russian).
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Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter.
SIAM J. Math. Anal. 21 (4), pp. 995–1018.
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Error analysis in a uniform asymptotic expansion for the generalised exponential integral.
J. Comput. Appl. Math. 80 (1), pp. 127–161.
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17: 13.27 Mathematical Applications
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►Vilenkin (1968, Chapter 8) constructs irreducible representations of this group, in which the diagonal matrices correspond to operators of multiplication by an exponential function.
The other group elements correspond to integral operators whose kernels can be expressed in terms of Whittaker functions.
This identification can be used to obtain various properties of the Whittaker functions, including recurrence relations and derivatives.
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18: 6.18 Methods of Computation
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►The attainable accuracy of the asymptotic expansions can be increased considerably by exponential improvement.
Also, other ranges of can be covered by use of the continuation formulas of §6.4.
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►For example, the Gauss–Laguerre formula (§3.5(v)) can be applied to (6.2.2); see Todd (1954) and Tseng and Lee (1998).
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►Lastly, the continued fraction (6.9.1) can be used if is bounded away from the origin.
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, , and can be computed by Miller’s algorithm (§3.6(iii)), starting with initial values , say, where is an arbitrary large integer, and normalizing via .
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19: 19.38 Approximations
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►Minimax polynomial approximations (§3.11(i)) for and in terms of with can be found in Abramowitz and Stegun (1964, §17.3) with maximum absolute errors ranging from 4×10⁻⁵ to 2×10⁻⁸.
Approximations of the same type for and for are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸.
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►The accuracy is controlled by the number of terms retained in the approximation; for real variables the number of significant figures appears to be roughly twice the number of terms retained, perhaps even for near with the improvements made in the 1970 reference.
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