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21: Bibliography B

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Transcendental Functions Satisfying Nonhomogeneous Linear Differential Equations. The Macmillan Co., New York.

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External Links:

MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

§11.4(i), §11.4(v), §11.5(ii), §11.5(i), §11.5(ii), §11.5(iii), §11.7(ii), §11.7(iii), §11.7(v), §11.9(i), §11.9(iv), §11.9(iv), Ch.11, §14.29.

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

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Notes:

Unaltered republication of the original 1927 Harvard University edition.

External Links:

MathReview, ZentralBlatt

Referenced by:

§8.1, Notations P, Notations Q.

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Discrete ordinate solution of Fokker-Planck equations with non-linear coefficients. Phys. Rev. A 31 (3), pp. 1855–1868.

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Referenced by:

§18.39(iii).

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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

Referenced by:

§3.2(iii).

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The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods. John Wiley & Sons Ltd., Chichester.

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External Links:

ISBN 0-471-91046-5, MathReview (Peter Alfeld), ZentralBlatt

Referenced by:

§3.7(v).

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22: 1.16 Distributions

… ►The linear space of all test functions with the above definition of convergence is called a test function space. … ►Tempered distributions are continuous linear functionals on this space of test functions. … ►

§1.16(vii) Fourier Transforms of Tempered Distributions

… ►Then its Fourier transform is … ►

§1.16(viii) Fourier Transforms of Special Distributions

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23: Bibliography J

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A note on sampling expansion for a transform with parabolic cylinder kernel. Inform. Sci. 26 (2), pp. 155–158.

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ISSN 0020-0255, Document, MathReview, ZentralBlatt

Referenced by:

§12.16.

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Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II. Phys. D 2 (3), pp. 407–448.

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ISSN 0167-2789, Document, MathReview (V. A. Golubeva), ZentralBlatt

Referenced by:

§32.4(i), §32.4(ii), §32.4(iv), §32.4(v), §32.6(ii), §32.6(iii), §32.6(iv), §32.6(v), §32.6(v), §32.6(v).

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Fast Hankel transforms. Geophysical Prospecting 27 (4), pp. 876–901.

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External Links:

Document

Referenced by:

§10.74(vii).

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Asymptotic behavior of the continued fraction coefficients of a class of Stieltjes transforms including the Binet function. In Orthogonal functions, moment theory, and continued fractions (Campinas, 1996), Lecture Notes in Pure and Appl. Math., Vol. 199, pp. 257–274.

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External Links:

MathReview (Olav Njåstad), ZentralBlatt

Referenced by:

§5.10, §5.10.

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24: Bibliography T

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LSFBTR: A subroutine for calculating spherical Bessel transforms. Comput. Phys. Comm. 30 (1), pp. 93–99.

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Notes:

Single-precision Fortran.

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ISSN 0010-4655, Document, Code

Referenced by:

9th item.

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Laplace type integrals: Transformation to standard form and uniform asymptotic expansions. Quart. Appl. Math. 43 (1), pp. 103–123.

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ISSN 0033-569X, FullText, MathReview (I. Fenyő), ZentralBlatt

Referenced by:

§2.4(i).

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Numerical Linear Algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:

ISBN 0-89871-361-7, MathReview (Moody T. Chu), ZentralBlatt

Referenced by:

§3.2(iii), §3.2(vii).

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Algebraic transformations of hypergeometric functions and automorphic forms on Shimura curves. Trans. Amer. Math. Soc. 365 (12), pp. 6697–6729.

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ISSN 0002-9947, Document, FullText, MathReview (John M. Voight), ZentralBlatt

Referenced by:

§15.8(v), §15.8(v).

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Asymptotic solution of a linear nonhomogeneous second order differential equation with a transition point and its application to the computations of toroidal shells and propeller blades. J. Appl. Math. Mech. 23, pp. 1549–1565.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§9.1, Notations E, Notations E.

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25: 32.2 Differential Equations

… ►They are distinct modulo Möbius (bilinear) transformations …The fifty equations can be reduced to linear equations, solved in terms of elliptic functions (Chapters 22 and 23), or reduced to one of ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$. … ►

32.2.25 $$w(z;\alpha )=ϵW(\zeta )+\frac{1}{{ϵ}^5},$$

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Symbols:

$z$: real, $\alpha$: arbitrary constant and $W(\zeta )$: bilinear transformation

Permalink:

http://dlmf.nist.gov/32.2.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §32.2(vi), §32.2 and Ch.32

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32.2.27 $$\frac{d^{2}W}{{d\zeta }^2}=6W^2+\zeta +{ϵ}^6(2W^3+\zeta W);$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $W(\zeta )$: bilinear transformation

Permalink:

http://dlmf.nist.gov/32.2.E27

Encodings:

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See also:

Annotations for §32.2(vi), §32.2 and Ch.32

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32.2.28 $$w(z;\alpha ,\beta ,\gamma ,\delta )=1+2ϵW(\zeta ;a),$$

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Symbols:

$z$: real, $\alpha$: arbitrary constant, $\beta$: arbitrary constant, $\gamma$: arbitrary constant, $\delta$: arbitrary constant and $W(\zeta )$: bilinear transformation

Permalink:

http://dlmf.nist.gov/32.2.E28

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See also:

Annotations for §32.2(vi), §32.2 and Ch.32

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26: Bibliography O

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Hyperasymptotic solutions of second-order linear differential equations. I. Methods Appl. Anal. 2 (2), pp. 173–197.

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External Links:

ISSN 1073-2772, FullText, MathReview (A. D. Wood), ZentralBlatt

Referenced by:

§10.17(v), §10.40(iv), §13.29(i), §13.7(iii), §2.11(v), §9.7(v).

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On the calculation of Stokes multipliers for linear differential equations of the second order. Methods Appl. Anal. 2 (3), pp. 348–367.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Archibald D. MacGillivray), ZentralBlatt

Referenced by:

§2.7(ii).

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On the asymptotic and numerical solution of linear ordinary differential equations. SIAM Rev. 40 (3), pp. 463–495.

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ISSN 1095-7200, Document, MathReview (W. Balser), ZentralBlatt

Referenced by:

§16.25, §2.7(ii), §3.7(iii).

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Hyperasymptotic solutions of second-order linear differential equations. II. Methods Appl. Anal. 2 (2), pp. 198–211.

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External Links:

ISSN 1073-2772, FullText, MathReview (A. D. Wood), ZentralBlatt

Referenced by:

§10.17(v), §10.40(iv), §2.11(v), §9.7(v).

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Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one. Proc. Roy. Soc. London Ser. A 454, pp. 1–29.

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ISSN 1364-5021, Document, MathReview (Donald A. Lutz), ZentralBlatt

Referenced by:

§2.11(v), §2.7(ii).

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27: Bibliography L

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Optimal cylindrical and spherical Bessel transforms satisfying bound state boundary conditions. Comput. Phys. Comm. 99 (2-3), pp. 297–306.

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Notes:

Single-precision Fortran, maximum accuracy 8S.

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ISSN 0010-4655, Document, Code, ZentralBlatt

Referenced by:

4th item.

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Asymptotics of the first Appell function $F_1$ with large parameters II. Integral Transforms Spec. Funct. 24 (12), pp. 982–999.

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External Links:

ISSN 1065-2469, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§16.13, §16.13.

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Numerical Solution of Linear Difference Equations. NBSIR Technical Report 80-1976, National Bureau of Standards, Gaithersburg, MD 20899.

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Notes:

(Available from National Technical Information Service, Springfield, VA 22161.)

Referenced by:

§16.25, §3.6(vii).

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Bessel transforms and rational extrapolation. Numer. Math. 47 (1), pp. 1–14.

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External Links:

ISSN 0029-599X, Document, MathReview (Vítĕzslav Veselý), ZentralBlatt

Referenced by:

§10.74(vii).

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Linear prediction of bandlimited processes with flat spectral densities. IEEE Trans. Signal Process. 49 (7), pp. 1564–1569.

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External Links:

ISSN 1053-587X, Document, MathReview, ZentralBlatt

Referenced by:

§30.15(v).

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28: Bibliography F

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Un théorème de Paley-Wiener pour la transformation de Fourier sur un espace riemannien symétrique de rang un. J. Funct. Anal. 49 (2), pp. 230–268.

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External Links:

ISSN 0022-1236, Document, Link, MathReview (Mogens Flensted-Jensen), ZentralBlatt

Referenced by:

§18.39(i).

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Calculation of elliptic integrals of the third kind by means of Gauss’ transformation. Math. Comp. 19 (89), pp. 97–104.

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External Links:

FullText, MathReview (J. E. L. Peck), ZentralBlatt

Referenced by:

§19.36(ii).

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Numerical calculation of singular integrals related to Hankel transform. Comput. Math. Appl. 21 (2-3), pp. 87–94.

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Notes:

Fortran, minimum precision 6D.

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ISSN 0898-1221, Document, MathReview, ZentralBlatt

Referenced by:

§10.77(ix).

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On the asymptotic expansion of Mellin transforms. SIAM J. Math. Anal. 18 (1), pp. 273–282.

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External Links:

ISSN 0036-1410, Document, Link, MathReview (Archibald Brown), ZentralBlatt

Referenced by:

§2.3(vi).

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Über lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singulären Stellen. Math. Ann. 63 (3), pp. 301–321.

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ISSN 0025-5831, Document, ZentralBlatt

Referenced by:

§32.2(iv).

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29: Bibliography G

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The solution of Cauchy’s problem for two totally hyperbolic linear differential equations by means of Riesz integrals. Ann. of Math. (2) 48 (4), pp. 785–826.

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External Links:

FullText, MathReview (E. T. Copson), ZentralBlatt

Referenced by:

§35.2, §35.3(ii).

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The Computation of Special Functions by Linear Difference Equations. In Advances in Difference Equations (Veszprém, 1995), S. Elaydi, I. Győri, and G. Ladas (Eds.), pp. 213–243.

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External Links:

ISBN 90-5699-521-9, MathReview, ZentralBlatt

Referenced by:

§3.6(iii), §3.6(vii).

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Computing the zeros and turning points of solutions of second order homogeneous linear ODEs. SIAM J. Numer. Anal. 41 (3), pp. 827–855.

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External Links:

ISSN 0036-1429, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vi), §3.8(v).

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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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External Links:

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

Referenced by:

§15.17(v).

Encodings:

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

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Referenced by:

§18.39(iii).

Encodings:

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30: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

§1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ► $V$ becomes a normed linear vector space. … ►

Bounded and Unbounded Linear Operators

… ► … ► …