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1: 19.2 Definitions
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►where is a polynomial in while and are rational functions of .
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►Here are real parameters, and and are real or complex variables, with , .
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►If , then is pure imaginary.
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§19.2(iv) A Related Function:
… ►For the special cases of and see (19.6.15). …2: Bibliography G
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On high precision methods for computing integrals involving Bessel functions.
Math. Comp. 33 (147), pp. 1049–1057.
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The triplets of helium.
Philos. Trans. Roy. Soc. London Ser. A 228, pp. 151–196.
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Some integrals involving three modified Bessel functions. I.
J. Math. Phys. 27 (3), pp. 682–687.
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Matrix Computations.
3rd edition, Johns Hopkins University Press, Baltimore, MD.
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Constructing wavefunctions for nonlocal potentials.
J. Chem. Phys. 52, pp. 6211–6217.
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3: Bibliography S
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The Laplace Transform: Theory and Applications.
Undergraduate Texts in Mathematics, Springer-Verlag, New York.
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The determination of phases of wave functions.
Proc. Phys. Soc. 79 (6), pp. 1296–1297.
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The accuracy of iterated JWBK approximations for Coulomb radial functions.
Comput. Phys. Comm. 32 (2), pp. 115–119.
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Numerical evaluation of the Hankel transform.
Comput. Phys. Comm. 116 (2-3), pp. 278–294.
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Structure of avoided crossings for eigenvalues related to equations of Heun’s class.
J. Phys. A 30 (2), pp. 673–687.
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4: Bibliography D
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Integral Transforms and their Applications.
2nd edition, Applied Mathematical Sciences, Vol. 25, Springer-Verlag, New York.
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Methods of Numerical Integration.
2nd edition, Computer Science and Applied Mathematics, Academic Press Inc., Orlando, FL.
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Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann.
Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).
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Real zeros of hypergeometric polynomials.
J. Comput. Appl. Math. 247, pp. 152–161.
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Bessel functions and of integer order and complex argument.
Comput. Phys. Comm. 78 (1-2), pp. 181–189.
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5: 25.21 Software
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§25.21(vii) Fermi–Dirac and Bose–Einstein Integrals
…6: Bibliography F
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Tablicy značeniĭ funkcii ot kompleksnogo argumenta.
Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow (Russian).
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Application of the -function theory of Painlevé equations to random matrices: , , the LUE, JUE, and CUE.
Comm. Pure Appl. Math. 55 (6), pp. 679–727.
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Application of the -function theory of Painlevé equations to random matrices: , the JUE, CyUE, cJUE and scaled limits.
Nagoya Math. J. 174, pp. 29–114.
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Error bounds for a uniform asymptotic expansion of the Legendre function
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SIAM J. Math. Anal. 21 (2), pp. 523–535.
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Fast computation of incomplete elliptic integral of first kind by half argument transformation.
Numer. Math. 116 (4), pp. 687–719.
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7: 8.28 Software
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§8.28(vii) Generalized Exponential Integral for Complex Argument and/or Parameter
…8: 10.74 Methods of Computation
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►In the case of the modified Bessel function see especially Temme (1975).
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►It should be noted, however, that there is a difficulty in evaluating the coefficients , , , and , from the explicit expressions (10.20.10)–(10.20.13) when is close to owing to severe cancellation.
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►Similarly, to maintain stability in the interval the integration direction has to be forwards in the case of and backwards in the case of , with initial values obtained in an analogous manner to those for and .
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►Then and can be generated by either forward or backward recurrence on when , but if then to maintain stability has to be generated by backward recurrence on , and has to be generated by forward recurrence on .
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§10.74(vii) Integrals
…9: 33.23 Methods of Computation
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►§33.8 supplies continued fractions for and .
Combined with the Wronskians (33.2.12), the values of , , and their derivatives can be extracted.
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►Bardin et al. (1972) describes ten different methods for the calculation of and , valid in different regions of the ()-plane.
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§33.23(vii) WKBJ Approximations
… ►Hull and Breit (1959) and Barnett (1981b) give WKBJ approximations for and in the region inside the turning point: .10: 18.30 Associated OP’s
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►where is given by (18.30.2) and (18.30.3), with , , and as in (18.9.2).
…where the generalized hypergeometric function is defined by (16.2.1).
►For corresponding corecursive associated Jacobi polynomials, corecursive associated polynomials being discussed in §18.30(vii), see Letessier (1995).
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and of (18.30.23) and (18.30.24) are, also, precisely those of (18.2.34) and (18.2.35), now expressed via the traditional, , , coefficients, rather than the monic, , , recursion coefficients.
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