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11: Errata

… ►The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions. …

12: 19.29 Reduction of General Elliptic Integrals

… ► $I(𝐦)$ can be reduced to a linear combination of basic integrals and algebraic functions. …

13: 18.38 Mathematical Applications

… ►If we consider this abstract algebra with additional relation (18.38.9) and with dependence on $a,b,c,d$ according to (18.38.7) then it is isomorphic with the algebra generated by $K_0=L$ given by (18.28.6_2), $(K_{1}f)(z)=(z+z^{-1})f(z)$ and $K_2$ given by (18.38.4), and $K_0,K_1,K_2$ act on the linear span of the Askey–Wilson polynomials (18.28.1). …

14: DLMF Project News

error generating summary

15: 19.14 Reduction of General Elliptic Integrals

… ►Legendre (1825–1832) showed that every elliptic integral can be expressed in terms of the three integrals in (19.1.2) supplemented by algebraic, logarithmic, and trigonometric functions. …The last reference gives a clear summary of the various steps involving linear fractional transformations, partial-fraction decomposition, and recurrence relations. It then improves the classical method by first applying Hermite reduction to (19.2.3) to arrive at integrands without multiple poles and uses implicit full partial-fraction decomposition and implicit root finding to minimize computing with algebraic extensions. …

16: Bibliography G

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The GAP Group, Centre for Interdisciplinary Research in Computational Algebra, University of St. Andrews, United Kingdom.

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Groups, Algorithms, Programming: A system for computational discrete algebra with emphasis on computational group theory.

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GAP

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2nd item.

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Riemann surfaces, plane algebraic curves and their period matrices. J. Symbolic Comput. 26 (6), pp. 789–803.

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ISSN 0747-7171, Document, MathReview (Gabino González Díez), ZentralBlatt

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1st item.

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Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt

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§18.38(iii).

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Linear Differential Equations and Group Theory from Riemann to Poincaré. 2nd edition, Birkhäuser Boston Inc., Boston, MA.

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ISBN 0-8176-3837-7, MathReview, ZentralBlatt

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§15.17(v).

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Kinetic theory of linear shear flow. Phys. Fluids 1 (3), pp. 215–224.

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§18.39(iii).

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17: 15.19 Methods of Computation

… ►For $z\in ℝ$ it is always possible to apply one of the linear transformations in §15.8(i) in such a way that the hypergeometric function is expressed in terms of hypergeometric functions with an argument in the interval $[0,\frac{1}{2}]$. ►For $z\in ℂ$ it is possible to use the linear transformations in such a way that the new arguments lie within the unit circle, except when $z={\mathrm{e}}^{\pm \pi \mathrm{i}/3}$. … ►However, since the growth near the singularities of the differential equation is algebraic rather than exponential, the resulting instabilities in the numerical integration might be tolerable in some cases. … ►When $\mathrm{\Re }z>\frac{1}{2}$ it is better to begin with one of the linear transformations (15.8.4), (15.8.7), or (15.8.8). … …

18: Bibliography D

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Complex zeros of linear combinations of spherical Bessel functions and their derivatives. SIAM J. Math. Anal. 4 (1), pp. 128–133.

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ISSN 1095-7154, Document, MathReview (C. J. Bouwkamp), ZentralBlatt

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§10.58.

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Computing spectra of linear operators using the Floquet-Fourier-Hill method. J. Comput. Phys. 219 (1), pp. 296–321.

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ISSN 0021-9991, MathReview, ZentralBlatt

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§22.11.

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Computing Riemann matrices of algebraic curves. Phys. D 152/153, pp. 28–46.

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ISSN 0167-2789, Document, MathReview (John B. Little), ZentralBlatt

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3rd item.

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Abstract Algebra. 2nd edition, Prentice Hall Inc., Englewood Cliffs, N.J..

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ISBN 0-13-569302-0, MathReview (N. Popescu), ZentralBlatt

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§1.11(i), §1.11(ii), §1.11(iii).

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Convergent expansions for solutions of linear ordinary differential equations having a simple turning point, with an application to Bessel functions. Stud. Appl. Math. 107 (3), pp. 293–323.

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ISSN 0022-2526, Document, MathReview, ZentralBlatt

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§10.20(ii), §2.8(i).

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19: Bibliography B

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Algebraic Combinatorics. I: Association Schemes. The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA.

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ISBN 0-8053-0490-8, MathReview Entry, ZentralBlatt

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§18.28(xi).

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Orthogonal Polynomials in Coding Theory and Algebraic Combinatorics. In Orthogonal Polynomials (Columbus, OH, 1989), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 294, pp. 25–53.

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MathReview, ZentralBlatt

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§18.38(iii).

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An Introduction to Linear Difference Equations. Dover Publications Inc., New York.

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Unaltered republication of the original 1927 Harvard University edition.

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MathReview, ZentralBlatt

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§8.1, Notations P, Notations Q.

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Error estimates for the solution of linear systems. SIAM J. Sci. Comput. 21 (2), pp. 764–781.

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ISSN 1095-7197, Document, MathReview (Dimitrios Noutsos), ZentralBlatt

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§3.2(iii).

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Plane Algebraic Curves. Birkhäuser Verlag, Basel.

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Translated from the German by John Stillwell

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ISBN 3-7643-1769-8, MathReview, ZentralBlatt

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§21.7(i).

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20: Bibliography W

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Linear difference equations with transition points. Math. Comp. 74 (250), pp. 629–653.

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ISSN 0025-5718, Document, MathReview (B. G. Pachpatte), ZentralBlatt

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§2.9(iii).

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Linear Turning Point Theory. Applied Mathematical Sciences No. 54, Springer-Verlag, New York.

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ISBN 0-387-96046-5, MathReview (Anthony Leung), ZentralBlatt

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Ch.9.

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Linear and Nonlinear Waves. John Wiley & Sons, New York.

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Reprinted in 1999.

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ISBN 0471359424, MathReview, ZentralBlatt

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§9.16.

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The Algebraic Eigenvalue Problem. Monographs on Numerical Analysis. Oxford Science Publications, The Clarendon Press, Oxford University Press, Oxford.

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Paperback reprint of original edition, published in 1965 by Clarendon Press, Oxford.

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ISBN 0-19-853418-3, MathReview, ZentralBlatt

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§3.2(v), §3.2(vi), §3.2(vii).

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Asymptotics of linear recurrences. Anal. Appl. (Singap.) 12 (4), pp. 463–484.

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ISSN 0219-5305, Document, Link, MathReview Entry, ZentralBlatt

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§2.9(iii), §2.9(iii).

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