…
►Hence, if is fixed and , then , , or according as , , or ; compare (6.2.14).
…
►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.
…
►
…
►Complex values of the variables are allowed, with some restrictions in the case of that are sufficient but not always necessary.
…
►Accurate values of for near 0 can be obtained from by (19.2.6) and (19.25.13).
…
►This method loses significant figures in if and are nearly equal unless they are given exact values—as they can be for tables.
…
►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).
…
►When the values of complete integrals are known, addition theorems with (§19.11(ii)) ease the computation of functions such as when is small and positive.
…
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).
Olver (1960) tabulates
, ,
, , , , 8D.
Also included are tables of the coefficients in the uniform asymptotic
expansions of these zeros and associated values as .
…
►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 .
…
►Lastly, as ,
…uniformly for bounded real values of .
…
►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.
…
…
►
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.
…
►where the sum is over nonnegative integer values of for which .
…
►The quantity is real-valued.
…
…
►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 .
…
►In this way we arrive athyperasymptotic expansions.
…
►17408, compared with the correct value
…
►Comparison with the true value
…
►For example, using double precision is found to agree with (2.11.31) to 13D.
…
…
►If , then has no positive real zeros, and if , , then has a zero at
.
…
►When , has a string of complex zeros that approaches the ray as , and a conjugate string.
…
►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.
…
►
…
►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
.
The principal branch has a cut along the interval and agrees with (25.12.1) when ; see also §4.2(i).
…
►
…
►For other values of , is defined by analytic continuation.
…
The asymptotic results were originally for real valued and .
However, these results are also valid for complex values of . The maximum sectors of validity are
now specified.
A new paragraph with several new equations
and a new reference has been added at the end
to provide asymptotic expansions for
Kummer functions and as
in and and fixed.