large argument
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31—40 of 44 matching pages
31: 15.19 Methods of Computation
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►For it is always possible to apply one of the linear transformations in §15.8(i) in such a way that the hypergeometric function is expressed in terms of hypergeometric functions with an argument in the interval .
►For it is possible to use the linear transformations in such a way that the new arguments lie within the unit circle, except when .
This is because the linear transformations map the pair onto itself.
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►Large values of or , for example, delay convergence of the Gauss series, and may also lead to severe cancellation.
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►For example, in the half-plane we can use (15.12.2) or (15.12.3) to compute and , where is a large positive integer, and then apply (15.5.18) in the backward direction.
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32: Bibliography D
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Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter.
SIAM J. Math. Anal. 21 (4), pp. 995–1018.
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33: 10.75 Tables
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►Also, for additional listings of tables pertaining to complex arguments see Babushkina et al. (1997).
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British Association for the Advancement of Science (1937) tabulates , , , 10D; , , , 8–9S or 8D. Also included are auxiliary functions to facilitate interpolation of the tables of , for small values of , as well as auxiliary functions to compute all four functions for large values of .
34: Bibliography O
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Uniform asymptotic expansions for hypergeometric functions with large parameters. I.
Analysis and Applications (Singapore) 1 (1), pp. 111–120.
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Uniform asymptotic expansions for hypergeometric functions with large parameters. II.
Analysis and Applications (Singapore) 1 (1), pp. 121–128.
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Uniform asymptotic expansions for hypergeometric functions with large parameters. III.
Analysis and Applications (Singapore) 8 (2), pp. 199–210.
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A class of integral identities with matrix argument.
Duke Math. J. 26 (2), pp. 207–213.
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Some new asymptotic expansions for Bessel functions of large orders.
Proc. Cambridge Philos. Soc. 48 (3), pp. 414–427.
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35: 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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Computing solutions of the modified Bessel differential equation for imaginary orders and positive arguments.
ACM Trans. Math. Software 30 (2), pp. 145–158.
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36: Bibliography S
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Parabolic cylinder functions of integer and half-integer orders for nonnegative arguments.
Comput. Phys. Comm. 115 (1), pp. 69–86.
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Inequalities involving cylindrical functions of nearly equal argument and order.
Proc. Amer. Math. Soc. 5 (3), pp. 337–344.
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An inequality involving Bessel functions of argument nearly equal to their order.
Proc. Amer. Math. Soc. 4 (6), pp. 858–859.
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Large orders and summability of eigenvalue perturbation theory: A mathematical overview.
Int. J. Quantum Chem. 21, pp. 3–25.
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37: 4.45 Methods of Computation
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►Another method, when is large, is to sum
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►The inverses , , and can be computed from the logarithmic forms given in §4.37(iv), with real arguments.
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►Initial approximations are obtainable, for example, from the power series (4.13.6) (with ) when is close to , from the asymptotic expansion (4.13.10) when is large, and by numerical integration of the differential equation (4.13.4) (§3.7) for other values of .
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38: Bibliography K
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Approximation Formulae for Generalized Hypergeometric Functions for Large Values of the Parameters.
J. B. Wolters, Groningen.
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Algorithm 877: A subroutine package for cylindrical functions of complex order and nonnegative argument.
ACM Trans. Math. Software 34 (4), pp. Art. 22, 21.
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Algorithm 912: a module for calculating cylindrical functions of complex order and complex argument.
ACM Trans. Math. Software 37 (4), pp. Art. 47, 25.
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The efficient evaluation of the hypergeometric function of a matrix argument.
Math. Comp. 75 (254), pp. 833–846.
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Programs for computing the logarithm of the gamma function, and the digamma function, for complex argument.
Comput. Phys. Comm. 4, pp. 221–226.
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39: 30.11 Radial Spheroidal Wave Functions
40: Bibliography B
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Anharmonic oscillator. II. A study of perturbation theory in large order.
Phys. Rev. D 7, pp. 1620–1636.
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Coulomb functions for large charges and small velocities.
Phys. Rev. (2) 97 (2), pp. 542–554.
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Table of characteristic values of Mathieu’s equation for large values of the parameter.
J. Washington Acad. Sci. 45 (6), pp. 166–196.
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Problem of two Coulomb centres at large intercentre separation: Asymptotic expansions from analytical solutions of the Heun equation.
J. Phys. A 30 (2), pp. 559–571.
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Generalized hypergeometric functions at unit argument.
Proc. Amer. Math. Soc. 114 (1), pp. 145–153.
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