Schl%C3%A4fli-type%20integrals
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21: 6.16 Mathematical Applications
§6.16(i) The Gibbs Phenomenon
… ►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. … ►§6.16(ii) Number-Theoretic Significance of
►If we assume Riemann’s hypothesis that all nonreal zeros of have real part of (§25.10(i)), then …22: 25.20 Approximations
Cody et al. (1971) gives rational approximations for in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are , , , . Precision is varied, with a maximum of 20S.
23: 23.14 Integrals
§23.14 Integrals
►24: 9.18 Tables
Zhang and Jin (1996, p. 337) tabulates , , , for to 8S and for to 9D.
Sherry (1959) tabulates , , , , ; 20S.
Zhang and Jin (1996, p. 339) tabulates , , , , , , , , ; 8D.
§9.18(v) Integrals
…25: 7.23 Tables
Abramowitz and Stegun (1964, Chapter 7) includes , , , 10D; , , 8S; , , 7D; , , , 6S; , , 10D; , , 9D; , , , 7D; , , , , 15D.
Abramowitz and Stegun (1964, Table 27.6) includes the Goodwin–Staton integral , , 4D; also , , 4D.
Zhang and Jin (1996, pp. 637, 639) includes , , , 8D; , , , 8D.
Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of , , , 7D and 8D, respectively; the real and imaginary parts of , , , 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.
Zhang and Jin (1996, p. 642) includes the first 10 zeros of , 9D; the first 25 distinct zeros of and , 8S.
26: 7.8 Inequalities
27: 19.36 Methods of Computation
§19.36 Methods of Computation
… ►Legendre’s integrals can be computed from symmetric integrals by using the relations in §19.25(i). … ►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). … ►Numerical quadrature is slower than most methods for the standard integrals but can be useful for elliptic integrals that have complicated representations in terms of standard integrals. … ►28: 11.14 Tables
Zhang and Jin (1996) tabulates and for and to 8D or 7S.
§11.14(iii) Integrals
►Abramowitz and Stegun (1964, Chapter 12) tabulates and for to 5D or 7D; , , and for to 6D.
Agrest et al. (1982) tabulates and for to 11D.