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11: 32.7 Bäcklund Transformations
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►satisfies with
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§32.7(vii) Sixth Painlevé Equation
►Let , , be solutions of with … ► also has quadratic and quartic transformations. …Also, …12: Bibliography Q
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“Best possible” upper and lower bounds for the zeros of the Bessel function
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Trans. Amer. Math. Soc. 351 (7), pp. 2833–2859.
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13: 8.17 Incomplete Beta Functions
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►For a historical profile of see Dutka (1981).
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►where
…The and convergents are less than , and the and convergents are greater than .
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►For or , more rapid convergence is obtained by computing and using (8.17.4).
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§8.17(vii) Addendum to 8.17(i) Definitions and Basic Properties
…14: Bibliography T
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Asymptotic estimates of Stirling numbers.
Stud. Appl. Math. 89 (3), pp. 233–243.
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High Speed Numerical Integration of Fermi Dirac Integrals.
Master’s Thesis, Naval Postgraduate School, Monterey, CA.
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The Theory of Functions.
2nd edition, Oxford University Press, Oxford.
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Numerical Linear Algebra.
Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
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Rational Chebyshev approximation for the Fermi-Dirac integral
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Solid–State Electronics 41 (5), pp. 771–773.
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15: 34.3 Basic Properties: Symbol
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►When any one of is equal to , or , the symbol has a simple algebraic form.
…For these and other results, and also cases in which any one of is or , see Edmonds (1974, pp. 125–127).
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►Even permutations of columns of a symbol leave it unchanged; odd permutations of columns produce a phase factor , for example,
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§34.3(vii) Relations to Legendre Polynomials and Spherical Harmonics
►For the polynomials see §18.3, and for the function see §14.30. …16: 12.10 Uniform Asymptotic Expansions for Large Parameter
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►These cases are treated in §§12.10(vii)–12.10(viii).
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►Higher polynomials can be calculated from the recurrence relation
…and the then follow from
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§12.10(vii) Negative , . Expansions in Terms of Airy Functions
… ►The coefficients and are given by …17: 28.4 Fourier Series
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§28.4(vii) Asymptotic Forms for Large
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28.4.24
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28.4.25
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28.4.26
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►For the basic solutions and see §28.2(ii).
18: 11.10 Anger–Weber Functions
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►The Anger function and Weber function are defined by
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►The associated Anger–Weber function is defined by
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§11.10(vii) Special Values
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11.10.26
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11.10.29
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19: 3.6 Linear Difference Equations
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►In practice, however, problems of severe instability often arise and in §§3.6(ii)–3.6(vii) we show how these difficulties may be overcome.
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►Given numerical values of and , the solution of the equation
…These errors have the effect of perturbing the solution by unwanted small multiples of and of an independent solution , say.
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►beginning with .
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