values%20at%20infinity
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1: 6.16 Mathematical Applications
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►Hence, if is fixed and , then , , or according as , , or ; compare (6.2.14).
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►The first maximum of for positive occurs at
and equals ; compare Figure 6.3.2.
Hence if and , then the limiting value of overshoots by approximately 18%.
Similarly if , then the limiting value of undershoots by approximately 10%, and so on.
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2: 19.36 Methods of Computation
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►Complex values of the variables are allowed, with some restrictions in the case of that are sufficient but not always necessary.
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►Accurate values of for near 0 can be obtained from by (19.2.6) and (19.25.13).
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►This method loses significant figures in if and are nearly equal unless they are given exact values—as they can be for tables.
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►The cases and require different treatment for numerical purposes, and again precautions are needed to avoid cancellations.
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►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20).
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3: 10.75 Tables
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Olver (1960) tabulates , , , , , , , , , , 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as ; see §10.21(viii), and more fully Olver (1954).
Makinouchi (1966) tabulates all values of and in the interval , with at least 29S. These are for , 10, 20; , ; with and , except for .
Zhang and Jin (1996, p. 270) tabulates , , , , , 8D.
Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of and , for , 9S.
Olver (1960) tabulates , , , , , , 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as .
4: 18.40 Methods of Computation
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►There are many ways to implement these first two steps, noting that the expressions for and of equation (18.2.30) are of little practical numerical value, see Gautschi (2004) and Golub and Meurant (2010).
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►Results of low ( to decimal digits) precision for are easily obtained for to .
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►The bottom and top of the steps at the are lower and upper bounds to as made explicit via the Chebyshev inequalities discussed by Shohat and Tamarkin (1970, pp. 42–43).
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►This is a challenging case as the desired on has an essential singularity at
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►Achieving precisions at this level shown above requires higher than normal computational precision, see Gautschi (2009).
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5: 5.11 Asymptotic Expansions
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►Wrench (1968) gives exact values of up to .
Spira (1971) corrects errors in Wrench’s results and also supplies exact and 45D values of for .
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►Lastly, as ,
…uniformly for bounded real values of .
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►If the sums in the expansions (5.11.1) and (5.11.2) are terminated at
() and is real and positive, then the remainder terms are bounded in magnitude by the first neglected terms and have the same sign.
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6: 26.10 Integer Partitions: Other Restrictions
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denotes the number of partitions of into at most distinct parts.
denotes the number of partitions of into parts with difference at least .
denotes the number of partitions of into parts with difference at least 3, except that multiples of 3 must differ by at least 6.
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►where the sum is over nonnegative integer values of for which .
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►The quantity is real-valued.
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7: 2.11 Remainder Terms; Stokes Phenomenon
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►If the results agree within significant figures, then it is likely—but not certain—that the truncated asymptotic series will yield at least correct significant figures for larger values of .
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►In this way we arrive at hyperasymptotic expansions.
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►17408, compared with the correct value
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►Comparison with the true value
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►For example, using double precision is found to agree with (2.11.31) to 13D.
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8: 12.11 Zeros
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►If , then has no positive real zeros, and if , , then has a zero at
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►When , has a string of complex zeros that approaches the ray as , and a conjugate string.
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►For large negative values of the real zeros of , , , and can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii).
…as () , fixed.
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12.11.9
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9: 25.12 Polylogarithms
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►Other notations and names for include (Kölbig et al. (1970)), Spence function (’t Hooft and Veltman (1979)), and (Maximon (2003)).
►In the complex plane has a branch point at
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The principal branch has a cut along the interval and agrees with (25.12.1) when ; see also §4.2(i).
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►For other values of , is defined by analytic continuation.
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10: Bibliography B
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Pionic atoms.
Annual Review of Nuclear and Particle Science 20, pp. 467–508.
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Coulomb functions (negative energies).
Comput. Phys. Comm. 20 (3), pp. 447–458.
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Some solutions of the problem of forced convection.
Philos. Mag. Series 7 20, pp. 322–343.
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Table of characteristic values of Mathieu’s equation for large values of the parameter.
J. Washington Acad. Sci. 45 (6), pp. 166–196.
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Generalized hypergeometric functions at unit argument.
Proc. Amer. Math. Soc. 114 (1), pp. 145–153.
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