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21: 3.4 Differentiation
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Two-Point Formula
… ►Three-Point Formula
… ►Four-Point Formula
… ►Five-Point Formula
… ►The choice is motivated by saddle-point analysis; see §2.4(iv) or examples in §3.5(ix). …22: 8 Incomplete Gamma and Related
Functions
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23: 28 Mathieu Functions and Hill’s Equation
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24: 8.26 Tables
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Khamis (1965) tabulates for , to 10D.
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.
25: 23 Weierstrass Elliptic and Modular
Functions
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26: Bibliography M
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Rational approximations, software and test methods for sine and cosine integrals.
Numer. Algorithms 12 (3-4), pp. 259–272.
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On a class of algebraic solutions to the Painlevé VI equation, its determinant formula and coalescence cascade.
Funkcial. Ekvac. 46 (1), pp. 121–171.
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Calculation of the modified Bessel functions of the second kind with complex argument.
Math. Comp. 20 (95), pp. 407–412.
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Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems.
Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.
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The -analogue of the Laguerre polynomials.
J. Math. Anal. Appl. 81 (1), pp. 20–47.
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27: 8.13 Zeros
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►As increases the positive zeros coalesce to form a double zero at ().
…When a pair of conjugate trajectories emanate from the point
in the complex -plane.
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28: 32.2 Differential Equations
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►An equation is said to have the Painlevé property if all its solutions are free from movable branch points; the solutions may have movable poles or movable isolated essential singularities (§1.10(iii)), however.
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§32.2(vi) Coalescence Cascade
► – are obtained from by a coalescence cascade: …29: Gergő Nemes
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►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions.
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30: Wolter Groenevelt
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►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials.
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