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11—18 of 18 matching pages
11: 9.18 Tables
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Miller (1946) tabulates , for , for ; , for ; , for ; , , , (respectively , , , ) for . Precision is generally 8D; slightly less for some of the auxiliary functions. Extracts from these tables are included in Abramowitz and Stegun (1964, Chapter 10), together with some auxiliary functions for large arguments.
Zhang and Jin (1996, p. 337) tabulates , , , for to 8S and for to 9D.
§9.18(iii) Complex Variables
… ►Sherry (1959) tabulates , , , , ; 20S.
12: Bibliography G
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Algorithm 259: Legendre functions for arguments larger than one.
Comm. ACM 8 (8), pp. 488–492.
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Some integrals involving three Bessel functions when their arguments satisfy the triangle inequalities.
J. Math. Phys. 25 (11), pp. 3350–3356.
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Evaluation of Legendre functions of argument greater than one.
Comput. Phys. Comm. 105 (2-3), pp. 273–283.
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Algorithm 831: Modified Bessel functions of imaginary order and positive argument.
ACM Trans. Math. Software 30 (2), pp. 159–164.
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Algorithm 939: computation of the Marcum Q-function.
ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
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13: 9.7 Asymptotic Expansions
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9.7.1
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9.7.3
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9.7.4
►Numerical values of are given in Table 9.7.1 for to 2D.
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§9.7(iv) Error Bounds for Complex Variables
…14: Bibliography C
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Bessel functions and of real order and complex argument.
Comput. Phys. Comm. 24 (1), pp. 97–105.
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Asymptotic estimates for generalized Stirling numbers.
Analysis (Munich) 20 (1), pp. 1–13.
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Validated computation of certain hypergeometric functions.
ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
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Numerical evaluation of Airy functions with complex arguments.
J. Comput. Phys. 99 (1), pp. 106–114.
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Tables of Jacobian Elliptic Functions Whose Arguments are Rational Fractions of the Quarter Period.
National Physical Laboratory Mathematical Tables, Vol. 7, Her Majesty’s Stationery Office, London.
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15: 22.3 Graphics
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, , and as functions of real arguments
and .
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§22.3(iii) Complex ; Real
… ►§22.3(iv) Complex
… ► … ► …16: 20.11 Generalizations and Analogs
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►If both are positive, then allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)):
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►where and .
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20.11.4
►In the case identities for theta functions become identities in the complex variable , with , that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7).
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►However, in this case is no longer regarded as an independent complex variable within the unit circle, because is related to the variable of the theta functions via (20.9.2).
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17: Bibliography R
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On the definition and properties of generalized - symbols.
J. Math. Phys. 20 (12), pp. 2398–2415.
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Computation of Hankel (Bessel) functions of complex index and argument by numerical integration of a Schläfli contour integral.
Ž. Vyčisl. Mat. i Mat. Fiz. 13, pp. 1415–1424, 1636.
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Mathieu functions of integral orders and real arguments.
IEEE Trans. Microwave Theory Tech. 28 (3), pp. 276–277.
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Total positivity properties of generalized hypergeometric functions of matrix argument.
J. Statist. Phys. 116 (1-4), pp. 907–922.
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Real and complex analysis.
McGraw-Hill Book Co., New York-Toronto, Ont.-London.
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18: Bibliography W
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The Nahm equations, finite-gap potentials and Lamé functions.
J. Phys. A 20 (10), pp. 2679–2683.
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Asymptotic expansions of some matrix argument hypergeometric functions, with applications to macromolecules.
Ann. Inst. Statist. Math. 45 (3), pp. 467–475.
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Wave functions for large arguments by the amplitude-phase method.
Phys. Rev. 52, pp. 1123–1127.
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Uniform asymptotic expansion of the Jacobi polynomials in a complex domain.
Proc. Roy. Soc. London Ser. A 460, pp. 2569–2586.
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Four-figure tables of the Airy function in the complex plane.
Philos. Mag. (7) 37, pp. 236–261.
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