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11: 28.6 Expansions for Small
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►Leading terms of the power series for and for are:
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►The coefficients of the power series of , and also , are the same until the terms in and , respectively.
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►Numerical values of the radii of convergence of the power series (28.6.1)–(28.6.14) for are given in Table 28.6.1.
Here for , for , and for and .
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§28.6(ii) Functions and
…12: 10.75 Tables
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Achenbach (1986) tabulates , , , , , 20D or 18–20S.
Makinouchi (1966) tabulates all values of and in the interval , with at least 29S. These are for , 10, 20; , ; with and , except for .
Abramowitz and Stegun (1964, Chapter 11) tabulates , , , 10D; , , , 8D.
Leung and Ghaderpanah (1979), tabulates all zeros of the principal value of , for , 29S.
Abramowitz and Stegun (1964, Chapter 11) tabulates , , , 7D; , , , 6D.
13: 26.12 Plane Partitions
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26.12.9
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26.12.10
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26.12.11
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►The notation denotes the sum over all plane partitions contained in , and denotes the number of elements in .
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►where is the sum of the squares of the divisors of .
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14: 3.6 Linear Difference Equations
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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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►The unwanted multiples of now decay in comparison with , hence are of little consequence.
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►The latter method is usually superior when the true value of is zero or pathologically small.
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►beginning with .
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15: 5.10 Continued Fractions
16: Bibliography
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On the zeros of confluent hypergeometric functions. III. Characterization by means of nonlinear equations.
Lett. Nuovo Cimento (2) 29 (11), pp. 353–358.
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Uniform asymptotic expansions for exponential integrals and Bickley functions
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ACM Trans. Math. Software 9 (4), pp. 467–479.
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Special value of the hypergeometric function and connection formulae among asymptotic expansions.
J. Indian Math. Soc. (N.S.) 51, pp. 161–221.
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Normal forms of functions near degenerate critical points, the Weyl groups and Lagrangian singularities.
Funkcional. Anal. i Priložen. 6 (4), pp. 3–25 (Russian).
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Normal forms of functions in the neighborhood of degenerate critical points.
Uspehi Mat. Nauk 29 (2(176)), pp. 11–49 (Russian).
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17: 3.7 Ordinary Differential Equations
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►The path is partitioned at points labeled successively , with , .
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►Write , , expand and in Taylor series (§1.10(i)) centered at , and apply (3.7.2).
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►If, for example, , then on moving the contributions of and to the right-hand side of (3.7.13) the resulting system of equations is not tridiagonal, but can readily be made tridiagonal by annihilating the elements of that lie below the main diagonal and its two adjacent diagonals.
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►The values are the eigenvalues and the corresponding solutions of the differential equation are the eigenfunctions.
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►where and
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18: 24.20 Tables
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►Abramowitz and Stegun (1964, Chapter 23) includes exact values of , , ; , , , , 20D; , , 18D.
►Wagstaff (1978) gives complete prime factorizations of and for and , respectively.
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►For information on tables published before 1961 see Fletcher et al. (1962, v. 1, §4) and Lebedev and Fedorova (1960, Chapters 11 and 14).
19: 3.2 Linear Algebra
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►where , , , and
…Forward elimination for solving then becomes ,
…and back substitution is , followed by
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►Define the Lanczos vectors
and coefficients and by , a normalized vector (perhaps chosen randomly), , , and for by the recursive scheme
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►Start with , vector such that , , .
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20: 3.11 Approximation Techniques
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►Beginning with , , we apply
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►With , the last equations give as the solution of a system of linear equations.
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►(3.11.29) is a system of linear equations for the coefficients .
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►With this choice of and , the corresponding sum (3.11.32) vanishes.
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►Two are endpoints: and ; the other points and are control points.
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