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21: 35.10 Methods of Computation
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►See Yan (1992) for the and functions of matrix argument in the case
, and Bingham et al. (1992) for Monte Carlo simulation on applied to a generalization of the integral (35.5.8).
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22: 4.43 Cubic Equations
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4.43.2
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►Note that in Case (a) all the roots are real, whereas in Cases (b) and (c) there is one real root and a conjugate pair of complex roots.
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23: 31.12 Confluent Forms of Heun’s Equation
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►This has regular singularities at and , and an irregular singularity of rank 1 at .
►Mathieu functions (Chapter 28), spheroidal wave functions (Chapter 30), and Coulomb spheroidal functions (§30.12) are special cases of solutions of the confluent Heun equation.
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24: 10.42 Zeros
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►The distribution of the zeros of in the sector in the cases
is obtained on rotating Figures 10.21.2, 10.21.4, 10.21.6, respectively, through an angle so that in each case the cut lies along the positive imaginary axis.
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25: 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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►In the case of the spherical Bessel functions the explicit formulas given in §§10.49(i) and 10.49(ii) are terminating cases of the asymptotic expansions given in §§10.17(i) and 10.40(i) for the Bessel functions and modified Bessel functions.
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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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►In the case of , the need for initial values can be avoided by application of Olver’s algorithm (§3.6(v)) in conjunction with Equation (10.12.4) used as a normalizing condition, or in the case of noninteger orders, (10.23.15).
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►The spherical Bessel transform is the Hankel transform (10.22.76) in the case when is half an odd positive integer.
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26: 14.29 Generalizations
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►As in the case of (14.21.1), the solutions are hypergeometric functions, and (14.29.1) reduces to (14.21.1) when .
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27: 14.32 Methods of Computation
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►In other cases recurrence relations (§14.10) provide a powerful method when applied in a stable direction (§3.6); see Olver and Smith (1983) and Gautschi (1967).
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28: 29.11 Lamé Wave Equation
29: 30.2 Differential Equations
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§30.2(iii) Special Cases
…30: Bibliography L
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An underflow-induced graphics failure solved by SLI arithmetic.
In IEEE Symposium on Computer Arithmetic, E. E. Swartzlander, M. J. Irwin, and G. A. Jullien (Eds.),
Washington, D.C., pp. 10–17.
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