linear second order differential operator

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1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

§1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ► … ► … ► … ►

1.18.71

\mathcal{B}(f)=\lim_{x\to a+}\left(\alpha(x)f(x)+\beta(x)f^{\prime}(x)\right),

f\in\mathcal{D}({\mathcal{L}}^{*})

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Symbols:: ${\NVar{\mathbf{A}}}^{*}$ : adjoint of matrix and $\in$ : element of
Referenced by:: Erratum (V1.2.0) Section 1.18
Permalink:: http://dlmf.nist.gov/1.18.E71
Encodings:: pMML, png, TeX
See also:: Annotations for §1.18(ix), §1.18(ix), §1.18 and Ch.1

…

… ►This especially included updated information on matrix analysis, measure theory, spectral analysis, and a new section on linear second order differential operators and eigenfunction expansions. … ►The specific updates to Chapter 1 include the addition of an entirely new subsection §1.18 entitled “Linear Second Order Differential Operators and Eigenfunction Expansions” which is a survey of the formal spectral analysis of second order differential operators. … ►

Chapter 1 Additions

The following additions were made in Chapter 1:

Section 1.2

New subsections, 1.2(v) Matrices, Vectors, Scalar Products, and Norms and 1.2(vi) Square Matrices, with Equations (1.2.27)–(1.2.77).
Section 1.3

The title of this section was changed from “Determinants” to “Determinants, Linear Operators, and Spectral Expansions”. An extra paragraph just below (1.3.7). New subsection, 1.3(iv) Matrices as Linear Operators, with Equations (1.3.20), (1.3.21).
Section 1.4

In 1.4(v), three new paragraphs on Stieltjes integrals/measure, with Equations (1.4.23_1)–(1.4.23_3).
Section 1.8

In Subsection 1.8(i), the title of the paragraph “Bessel’s Inequality” was changed to “Parseval’s Formula”. We give the relation between the real and the complex coefficients, and include more general versions of Parseval’s Formula, Equations (1.8.6_1), (1.8.6_2). The title of Subsection 1.8(iv) was changed from “Transformations” to “Poisson’s Summation Formula”, and we added an extra remark just below (1.8.14).
Section 1.10

New subsection, 1.10(xi) Generating Functions, with Equations (1.10.26)–(1.10.29).
Section 1.13

New subsection, 1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms, with Equations (1.13.26)–(1.13.31).
Section 1.14(i)

Another form of Parseval’s formula, (1.14.7_5).
Section 1.16

We include several extra remarks and Equations (1.16.3_5), (1.16.9_5). New subsection, 1.16(ix) References for Section 1.16.
Section 1.17

Two extra paragraphs in Subsection 1.17(ii) Integral Representations, with Equations (1.17.12_1), (1.17.12_2); Subsection 1.17(iv) Mathematical Definitions is almost completely rewritten.
Section 1.18

An entire new section, 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions, including new subsections, 1.18(i)–1.18(x), and several equations, (1.18.1)–(1.18.71).

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4: 2.9 Difference Equations

… ►or equivalently the second-order homogeneous linear difference equation … ►This situation is analogous to second-order homogeneous linear differential equations with an irregular singularity of rank 1 at infinity (§2.7(ii)). … ►For asymptotic approximations to solutions of second-order difference equations analogous to the Liouville–Green (WKBJ) approximation for differential equations (§2.7(iii)) see Spigler and Vianello (1992, 1997) and Spigler et al. (1999). … ►For discussions of turning points, transition points, and uniform asymptotic expansions for solutions of linear difference equations of the second order see Wang and Wong (2003, 2005). ►For an introduction to, and references for, the general asymptotic theory of linear difference equations of arbitrary order, see Wimp (1984, Appendix B). …

5: Bibliography T

… ►

E. C. Titchmarsh (1946) Eigenfunction Expansions Associated with Second-Order Differential Equations. Clarendon Press, Oxford.

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External Links:: MathReview (H. Pollard)
Referenced by:: §1.18(x).
Encodings:: BibTeX

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E. C. Titchmarsh (1958) Eigenfunction Expansions Associated with Second Order Differential Equations, Part 2, Partial Differential Equations. Clarendon Press, Oxford.

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External Links:: ISBN 0198533187, MathReview Entry
Referenced by:: §1.18(x).
Encodings:: BibTeX

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E. C. Titchmarsh (1962a) Eigenfunction expansions associated with second-order differential equations. Part I. Second edition, Clarendon Press, Oxford.

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External Links:: MathReview Entry
Referenced by:: (1.18.59), §1.18(ix), §1.18(v), §1.18(vi), §1.18(x), (10.22.78), §10.22(v), §18.39(ii).
Encodings:: BibTeX

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L. N. Trefethen and D. Bau (1997) 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).
Encodings:: BibTeX

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S. A. Tumarkin (1959) 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.
Encodings:: BibTeX

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6: Bibliography D

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T. M. Dunster, D. A. Lutz, and R. Schäfke (1993) Convergent Liouville-Green expansions for second-order linear differential equations, with an application to Bessel functions. Proc. Roy. Soc. London Ser. A 440, pp. 37–54.

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External Links:: ISSN 0962-8444, FullText, MathReview (W. Balser), ZentralBlatt
Referenced by:: §10.41(iii), §2.8(i).
Encodings:: BibTeX

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T. M. Dunster (1990a) Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter. SIAM J. Math. Anal. 21 (4), pp. 995–1018.

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Notes:: Errata: In eq. (2.8) replace $(x^{2}/4)^{2}$ by $(x^{2}/4)^{s}$ . In eq. (4.2) $\phi(\zeta)$ should be replaced by $\psi(\zeta)$ . In the second line of eq. (4.7) insert an external factor $e^{-2\pi ij/3}$ and change the upper limit of the sum to $n-1$ . In eq. (4.15) the factor $\zeta$ is missing from the arguments of the functions Bi_ν and Bi’_ν.
External Links:: ISSN 0036-1410, Document, MathReview (A. de Castro Brzezicki), ZentralBlatt
Referenced by:: §10.24, §10.45, §2.8(iv).
Encodings:: BibTeX

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T. M. Dunster (1990b) Uniform asymptotic solutions of second-order linear differential equations having a double pole with complex exponent and a coalescing turning point. SIAM J. Math. Anal. 21 (6), pp. 1594–1618.

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External Links:: ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §14.26, §2.8(vi).
Encodings:: BibTeX

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T. M. Dunster (1994b) Uniform asymptotic solutions of second-order linear differential equations having a simple pole and a coalescing turning point in the complex plane. SIAM J. Math. Anal. 25 (2), pp. 322–353.

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External Links:: ISSN 0036-1410, Document, MathReview (H. C. Howard), ZentralBlatt
Referenced by:: §2.8(vi).
Encodings:: BibTeX

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T. M. Dunster (1996a) Asymptotic solutions of second-order linear differential equations having almost coalescent turning points, with an application to the incomplete gamma function. Proc. Roy. Soc. London Ser. A 452, pp. 1331–1349.

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External Links:: ISSN 0962-8444, FullText, MathReview, ZentralBlatt
Referenced by:: §2.8(vi), §8.12, §8.12.
Encodings:: BibTeX

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7: Bibliography K

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A. A. Kapaev (2004) Quasi-linear Stokes phenomenon for the Painlevé first equation. J. Phys. A 37 (46), pp. 11149–11167.

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External Links:: ISSN 0305-4470, Document, MathReview (W. Balser), ZentralBlatt
Referenced by:: §32.12(i).
Encodings:: BibTeX

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E. H. Kaufman and T. D. Lenker (1986) Linear convergence and the bisection algorithm. Amer. Math. Monthly 93 (1), pp. 48–51.

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External Links:: ISSN 0002-9890, FullText, MathReview (J. G. Herriot), ZentralBlatt
Referenced by:: §3.8(iii).
Encodings:: BibTeX

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T. H. Koornwinder (2006) Lowering and Raising Operators for Some Special Orthogonal Polynomials. In Jack, Hall-Littlewood and Macdonald Polynomials, Contemp. Math., Vol. 417, pp. 227–238.

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External Links:: FullText, MathReview Entry, ZentralBlatt
Referenced by:: §18.9(iii).
Encodings:: BibTeX

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J. J. Kovacic (1986) An algorithm for solving second order linear homogeneous differential equations. J. Symbolic Comput. 2 (1), pp. 3–43.

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External Links:: ISSN 0747-7171, MathReview, ZentralBlatt
Referenced by:: §31.14(ii).
Encodings:: BibTeX

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K. H. Kwon, L. L. Littlejohn, and G. J. Yoon (2006) Construction of differential operators having Bochner-Krall orthogonal polynomials as eigenfunctions. J. Math. Anal. Appl. 324 (1), pp. 285–303.

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External Links:: ISSN 0022-247X, Document, MathReview Entry, ZentralBlatt
Referenced by:: §18.36(i).
Encodings:: BibTeX

8: 18.27 $q$ -Hahn Class

… ►The $q$ -hypergeometric OP’s comprise the

q

-Hahn class (or

q

-linear lattice class) OP’s and the Askey–Wilson class (or

q

-quadratic lattice class) OP’s (§18.28). … ►The

q

-Hahn class OP’s comprise systems of OP’s

\{p_{n}(x)\}

n=0,1,\dots,N

, or

n=0,1,2,\dots

, that are eigenfunctions of a second order

q

-difference operator. …In the

q

-Hahn class OP’s the role of the operator

\ifrac{\mathrm{d}}{\mathrm{d}x}

in the Jacobi, Laguerre, and Hermite cases is played by the

q

-derivative

\mathcal{D}_{q}

, as defined in (17.2.41). … ►

18.27.16

\int_{0}^{\infty}L^{(\alpha)}_{n}\left(x;q\right)L^{(\alpha)}_{m}\left(x;q% \right)\frac{x^{\alpha}}{\left(-x;q\right)_{\infty}}\,\mathrm{d}x=\frac{\left(% q^{\alpha+1};q\right)_{n}}{\left(q;q\right)_{n}q^{n}}h_{0}^{(1)}\delta_{n,m},

\alpha>-1

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Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x};\NVar{q}\right)$ : $q$ -Laguerre polynomial, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable and $h_{n}$
Permalink:: http://dlmf.nist.gov/18.27.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §18.27(v), §18.27 and Ch.18

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18.27.19

\int_{0}^{\infty}\frac{S_{n}\left(x;q\right)S_{m}\left(x;q\right)}{\left(-x,-% qx^{-1};q\right)_{\infty}}\,\mathrm{d}x=\frac{\ln\left(q^{-1}\right)}{q^{n}}% \frac{\left(q;q\right)_{\infty}}{\left(q;q\right)_{n}}\delta_{n,m},

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Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $S_{\NVar{n}}\left(\NVar{x};\NVar{q}\right)$ : Stieltjes–Wigert polynomial, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$ : multiple $q$ -Pochhammer symbol, $q$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.27.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §18.27(vi), §18.27 and Ch.18

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9: Bibliography R

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M. Rahman (1981) A non-negative representation of the linearization coefficients of the product of Jacobi polynomials. Canad. J. Math. 33 (4), pp. 915–928.

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External Links:: ISSN 0008-414X, Document, Link, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.18(v).
Encodings:: BibTeX

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M. Reed and B. Simon (1978) Methods of Modern Mathematical Physics, Vol. 4, Analysis of Operators. Academic Press, New York.

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External Links:: ISBN 0-12-585004-2(vol.4), MathReview (P. R. Chernoff)
Referenced by:: §1.18(x).
Encodings:: BibTeX

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E. Ya. Remez (1957) General Computation Methods of Chebyshev Approximation. The Problems with Linear Real Parameters. Publishing House of the Academy of Science of the Ukrainian SSR, Kiev.

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Notes:: English translation, AEC-TR-4491, United States Atomic Energy Commission
External Links:: MathReview (A. S. Householder)
Referenced by:: §3.11(iii).
Encodings:: BibTeX

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S. Ritter (1998) On the computation of Lamé functions, of eigenvalues and eigenfunctions of some potential operators. Z. Angew. Math. Mech. 78 (1), pp. 66–72.

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External Links:: ISSN 0044-2267, Document, MathReview Entry, ZentralBlatt
Referenced by:: §29.20(ii).
Encodings:: BibTeX

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J. Rushchitsky and S. Rushchitska (2000) On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.

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External Links:: ISBN 3-7643-6287-1, MathReview Entry, ZentralBlatt
Referenced by:: 5th item.
Encodings:: BibTeX

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10: Bibliography J

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M. Jimbo and T. Miwa (1981) Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II. Phys. D 2 (3), pp. 407–448.

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External Links:: 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).
Encodings:: BibTeX

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D. S. Jones, M. J. Plank, and B. D. Sleeman (2010) Differential equations and mathematical biology. Chapman & Hall/CRC Mathematical and Computational Biology Series, CRC Press, Boca Raton, FL.

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Notes:: Second edition of Jones and Sleeman (2003)
External Links:: ISBN 978-1-4200-8357-6, MathReview Entry, ZentralBlatt
Referenced by:: Profile Brian D. Sleeman.
Encodings:: BibTeX

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D. S. Jones and B. D. Sleeman (2003) Differential equations and mathematical biology. Chapman & Hall/CRC Mathematical Biology and Medicine Series, Chapman & Hall/CRC, Boca Raton, FL.

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Notes:: A second edition was published in 2010.
External Links:: ISBN 1-58488-296-4, MathReview (Jian Hong Wu), ZentralBlatt
Referenced by:: D. S. Jones, M. J. Plank, and B. D. Sleeman (2010).
Encodings:: BibTeX

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D. S. Jones (2006) Parabolic cylinder functions of large order. J. Comput. Appl. Math. 190 (1-2), pp. 453–469.

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External Links:: ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt
Referenced by:: §12.10(vi).
Encodings:: BibTeX

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B. R. Judd (1998) Operator Techniques in Atomic Spectroscopy. Princeton University Press, Princeton, NJ.

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External Links:: ISBN 0-691-05901-2
Referenced by:: §34.12, Profile Brian R. Judd.
Encodings:: BibTeX

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linear second order differential operator

1—10 of 11 matching pages

1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

§1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

2: DLMF Project News

3: Errata

4: 2.9 Difference Equations

5: Bibliography T

6: Bibliography D

7: Bibliography K

8: 18.27 $q$ -Hahn Class

9: Bibliography R

10: Bibliography J

linear second order differential operator

1—10 of 11 matching pages

1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

§1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

2: DLMF Project News

3: Errata

4: 2.9 Difference Equations

5: Bibliography T

6: Bibliography D

7: Bibliography K

8: 18.27 q -Hahn Class

9: Bibliography R

10: Bibliography J

8: 18.27 $q$ -Hahn Class