Weber–Schafheitlin discontinuous integrals
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11: 10.22 Integrals
§10.22 Integrals
… ►Fractional Integral
… ►Weber–Schafheitlin Discontinuous Integrals, including Special Cases
… ►This is the Weber transform. … ►For collections of integrals of the functions , , , and , including integrals with respect to the order, see Andrews et al. (1999, pp. 216–225), Apelblat (1983, §12), Erdélyi et al. (1953b, §§7.7.1–7.7.7 and 7.14–7.14.2), Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2000, §§5.5 and 6.5–6.7), Gröbner and Hofreiter (1950, pp. 196–204), Luke (1962), Magnus et al. (1966, §3.8), Marichev (1983, pp. 191–216), Oberhettinger (1974, §§1.10 and 2.7), Oberhettinger (1990, §§1.13–1.16 and 2.13–2.16), Oberhettinger and Badii (1973, §§1.14 and 2.12), Okui (1974, 1975), Prudnikov et al. (1986b, §§1.8–1.10, 2.12–2.14, 3.2.4–3.2.7, 3.3.2, and 3.4.1), Prudnikov et al. (1992a, §§3.12–3.14), Prudnikov et al. (1992b, §§3.12–3.14), Watson (1944, Chapters 5, 12, 13, and 14), and Wheelon (1968).12: 10.75 Tables
British Association for the Advancement of Science (1937) tabulates , , , , , 10D; , , , , , 8D.
Wills et al. (1982) tabulates , , , for , 35D.
Zhang and Jin (1996, p. 199) tabulates the real and imaginary parts of the first 15 conjugate pairs of complex zeros of , , and the corresponding values of , , , respectively, 10D.
Abramowitz and Stegun (1964, Chapter 11) tabulates , , , 10D; , , , 8D.
Zhang and Jin (1996, p. 270) tabulates , , , , , 8D.
13: 11.14 Tables
Abramowitz and Stegun (1964, Chapter 12) tabulates and for to 5D or 7D; , , and for to 6D.
§11.14(iv) Anger–Weber Functions
►Bernard and Ishimaru (1962) tabulates and for and to 5D.
Jahnke and Emde (1945) tabulates for and to 4D.
§11.14(v) Incomplete Functions
…14: 11.16 Software
§11.16(iii) Integrals of Struve Functions
… ►§11.16(v) Anger and Weber Functions
… ►§11.16(vi) Integrals of Anger and Weber Functions
…15: 11.11 Asymptotic Expansions of Anger–Weber Functions
§11.11 Asymptotic Expansions of Anger–Weber Functions
►§11.11(i) Large , Fixed
… ► ►§11.11(ii) Large , Fixed
… ►(Note that Olver’s definition of omits the factor in (11.10.4).) …16: 10.21 Zeros
Zeros of and
… ►The first set of zeros of the principal value of are the points , , on the positive real axis (§10.21(i)). … ►The zeros of have a similar pattern to those of . …17: 11.13 Methods of Computation
§11.13(i) Introduction
… ►The treatment of Lommel and Anger–Weber functions is similar. … ►For numerical purposes the most convenient of the representations given in §11.5, at least for real variables, include the integrals (11.5.2)–(11.5.5) for and . …Other integrals that appear in §11.5(i) have highly oscillatory integrands unless is small. … ►See §3.6 for implementation of these methods, and with the Weber function as an example.18: 10.2 Definitions
Bessel Function of the Second Kind (Weber’s Function)
… ►Whether or not is an integer has a branch point at . … ►Except in the case of , the principal branches of and are two-valued and discontinuous on the cut ; compare §4.2(i). ►Both and are real when is real and . ►For fixed each branch of is entire in . …19: 10.15 Derivatives with Respect to Order
20: 11.15 Approximations
Luke (1975, pp. 416–421) gives Chebyshev-series expansions for , , , and , , for ; , , , and , , ; the coefficients are to 20D.
Newman (1984) gives polynomial approximations for for , , and rational-fraction approximations for for , . The maximum errors do not exceed 1.2×10⁻⁸ for the former and 2.5×10⁻⁸ for the latter.