Hankel%20loop%20integral
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21: 10.74 Methods of Computation
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►For evaluation of the Hankel functions and for complex values of and based on the integral representations (10.9.18) see Remenets (1973).
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§10.74(vi) Zeros and Associated Values
… ►Hankel Transform
… ►The spherical Bessel transform is the Hankel transform (10.22.76) in the case when is half an odd positive integer. …22: 10.54 Integral Representations
§10.54 Integral Representations
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10.54.1
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23: 10.11 Analytic Continuation
24: 10.20 Uniform Asymptotic Expansions for Large Order
§10.20 Uniform Asymptotic Expansions for Large Order
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10.20.6
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10.20.9
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§10.20(iii) Double Asymptotic Properties
►For asymptotic properties of the expansions (10.20.4)–(10.20.6) with respect to large values of see §10.41(v).25: 15.14 Integrals
§15.14 Integrals
… ►Integrals of the form and more complicated forms are given in Apelblat (1983, pp. 370–387), Prudnikov et al. (1990, §§1.15 and 2.21), Gradshteyn and Ryzhik (2000, §7.5) and Koornwinder (2015). … ►Hankel transforms of hypergeometric functions are given in Oberhettinger (1972, §1.17) and Erdélyi et al. (1954b, §8.17). ►For other integral transforms see Erdélyi et al. (1954b), Prudnikov et al. (1992b, §4.3.43), and also §15.9(ii).26: 8.26 Tables
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Khamis (1965) tabulates for , to 10D.
§8.26(iv) Generalized Exponential Integral
►Abramowitz and Stegun (1964, pp. 245–248) tabulates for , to 7D; also for , to 6S.
Pagurova (1961) tabulates for , to 4-9S; for , to 7D; for , to 7S or 7D.
Zhang and Jin (1996, Table 19.1) tabulates for , to 7D or 8S.
27: 10.2 Definitions
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Bessel Functions of the Third Kind (Hankel Functions)
►These solutions of (10.2.1) are denoted by and , and their defining properties are given by … ►The principal branches of and are two-valued and discontinuous on the cut . … ►Branch Conventions
…28: Bibliography N
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On an integral transform involving a class of Mathieu functions.
SIAM J. Math. Anal. 20 (6), pp. 1500–1513.
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Reduction and evaluation of elliptic integrals.
Math. Comp. 20 (94), pp. 223–231.
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Error Bounds for the Large-Argument Asymptotic Expansions of the Hankel and Bessel Functions.
Acta Appl. Math. 150, pp. 141–177.
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Elliptic integrals of the second and third kinds.
Zastos. Mat. 11, pp. 99–102.
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A table of integrals of the error functions.
J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
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29: Bibliography F
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Sur certaines sommes des intégral-cosinus.
Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
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Tables of Elliptic Integrals of the First, Second, and Third Kind.
Technical report
Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.
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Numerical calculation of singular integrals related to Hankel transform.
Comput. Math. Appl. 21 (2-3), pp. 87–94.
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A note on asymptotic evaluation of some Hankel transforms.
Math. Comp. 45 (172), pp. 537–548.
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