matrix notation

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21—30 of 30 matching pages

21: Bibliography D

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G. M. D’Ariano, C. Macchiavello, and M. G. A. Paris (1994) Detection of the density matrix through optical homodyne tomography without filtered back projection. Phys. Rev. A 50 (5), pp. 4298–4302.

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External Links:: Document
Referenced by:: §13.28(iii).
Encodings:: BibTeX

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H. T. Davis (1933) Tables of Higher Mathematical Functions I. Principia Press, Bloomington, Indiana.

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External Links:: ZentralBlatt
Referenced by:: §5.1, Notations P.
Encodings:: BibTeX

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P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou (1999b) Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math. 52 (11), pp. 1335–1425.

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External Links:: ISSN 0010-3640, Document, MathReview (D. S. Lubinsky), ZentralBlatt
Referenced by:: §18.32.
Encodings:: BibTeX

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R. B. Dingle (1973) Asymptotic Expansions: Their Derivation and Interpretation. Academic Press, London-New York.

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External Links:: MathReview, ZentralBlatt
Referenced by:: (11.11.11), §11.6(i), §11.6(iii), §2.11(v), Ch.2, §8.1, Notations, Notations.
Encodings:: BibTeX

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A. J. Durán and F. A. Grünbaum (2005) A survey on orthogonal matrix polynomials satisfying second order differential equations. J. Comput. Appl. Math. 178 (1-2), pp. 169–190.

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External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: §18.36(iv).
Encodings:: BibTeX

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22: Bibliography S

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G. W. Stewart (2001) Matrix Algorithms. Vol. 2: Eigensystems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:: ISBN 0-89871-503-2, MathReview Entry, ZentralBlatt
Referenced by:: §3.2(vi).
Encodings:: BibTeX

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J. A. Stratton, P. M. Morse, L. J. Chu, and R. A. Hutner (1941) Elliptic Cylinder and Spheroidal Wave Functions, Including Tables of Separation Constants and Coefficients. John Wiley and Sons, Inc., New York.

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External Links:: MathReview (H. Bateman), ZentralBlatt
Referenced by:: §28.1, 7th item, Notations S, Notations S.
Encodings:: BibTeX

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G. Szegő (1967) Orthogonal Polynomials. 3rd edition, American Mathematical Society, New York.

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Notes:: American Mathematical Society Colloquium Publications, Vol. 23
External Links:: MathReview (J. Burlak)
Referenced by:: Ch.14, §9.1, Notations A.
Encodings:: BibTeX

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G. Szegő (1975) Orthogonal Polynomials. 4th edition, Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, RI.

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External Links:: MathReview, ZentralBlatt
Referenced by:: §18.1(iii), §18.10(i), §18.10(ii), §18.10(iii), §18.11(ii), §18.12, §18.14(i), §18.14(iii), (18.15.4_5), §18.15(i), §18.15(ii), §18.15(ii), §18.15(ii), §18.15(iii), §18.15(iii), §18.15(iv), §18.15(v), §18.16(ii), §18.16(iv), §18.16(v), §18.16(vi), §18.17(viii), §18.18(i), §18.18(i), §18.18(viii), (18.2.1_5), (18.2.39), (18.2.4_5), §18.2(xi), §18.2(xii), §18.2(i), §18.2(iii), §18.2(iv), §18.2(v), Table 18.3.1, §18.31, §18.33(i), §18.33(ii), §18.33(iii), (18.35.9), §18.35(iii), §18.36(v), §18.5(i), §18.5(ii), §18.5(iii), §18.6(i), §18.7(i), §18.7(ii), §18.7(iii), §18.8, §18.9(i), §18.9(ii), §18.9(iii), Ch.18, Notations P.
Encodings:: BibTeX

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23: Bibliography I

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E. L. Ince (1940b) Further investigations into the periodic Lamé functions. Proc. Roy. Soc. Edinburgh 60, pp. 83–99.

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External Links:: MathReview (H. Bateman), ZentralBlatt
Referenced by:: §29.1, §29.17(ii), §29.3(iii), §29.6(i), §29.6(ii), §29.6(iii), §29.6(iv), Ch.29, Notations E, Notations E, Notations E, Notations E.
Encodings:: BibTeX

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M. E. H. Ismail and E. Koelink (2011) The

J

-matrix method. Adv. in Appl. Math. 46 (1-4), pp. 379–395.

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External Links:: ISSN 0196-8858, Document, Link, MathReview (Boris Petrovich Osilenker)
Referenced by:: §18.39(iv).
Encodings:: BibTeX

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24: Bibliography B

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P. M. Batchelder (1967) 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.
Encodings:: BibTeX

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C. Bingham, T. Chang, and D. Richards (1992) Approximating the matrix Fisher and Bingham distributions: Applications to spherical regression and Procrustes analysis. J. Multivariate Anal. 41 (2), pp. 314–337.

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External Links:: ISSN 0047-259X, Document, MathReview (A. L. Rukhin), ZentralBlatt
Referenced by:: §35.10, §35.9.
Encodings:: BibTeX

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P. Bleher and A. Its (1999) Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. of Math. (2) 150 (1), pp. 185–266.

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External Links:: ISSN 0003-486X, Document, MathReview (Thomas Kriecherbauer), ZentralBlatt
Referenced by:: §18.32.
Encodings:: BibTeX

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D. M. Bressoud (1999) Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture. Cambridge University Press, Cambridge.

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External Links:: ISBN 0-521-66170-6; 0-521-66646-5, MathReview (George E. Andrews), ZentralBlatt
Referenced by:: §1.3(ii), §26.10(ii), §26.10(iii), §26.12(i), §26.12(ii), §26.12(iii), §26.19, §26.20, §26.9(iii), Profile David M. Bressoud.
Encodings:: BibTeX

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J. T. Broad and W. P. Reinhardt (1976) One- and two-electron photoejection from

\mathrm{H}^{-}

: A multichannel

J

-matrix calculation. Phys. Rev. A 14, pp. 2159–2173.

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External Links:: Document, Link
Referenced by:: §18.39(iv).
Encodings:: BibTeX

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25: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Thus, in the notation of §1.17, we have an expansion … ►this being a matrix element of the resolvent

F(T)=\left(z-T\right)^{-1}

, this being a key quantity in many parts of physics and applied math, quantum scattering theory being a simple example, see Newton (2002, Ch. 7). … ►Note that the notations of (1.18.32) and (1.18.47) are used to distinguish the contributions from the discrete and continuous parts of the spectrum. … ► … ►In unusual cases

N=\infty

, even for all

\ell

, such as in the case of the Schrödinger–Coulomb problem (

V=-r^{-1}

) discussed in §18.39 and §33.14, where the point spectrum actually accumulates at the onset of the continuum at

\lambda=0

, implying an essential singularity, as well as a branch point, in matrix elements of the resolvent, (1.18.66). …

26: Bibliography K

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D. E. Knuth (1992) Two notes on notation. Amer. Math. Monthly 99 (5), pp. 403–422.

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External Links:: ISSN 0002-9890, FullText, MathReview (W. Dörfler), ZentralBlatt
Referenced by:: §26.1, §26.1, Notations, Notations.
Encodings:: BibTeX

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P. Koev and A. Edelman (2006) The efficient evaluation of the hypergeometric function of a matrix argument. Math. Comp. 75 (254), pp. 833–846.

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Notes:: For Matlab software for this function, see Plamen Koev’s home page.
External Links:: Home Page, ISSN 0025-5718, Document, MathReview (Luis Verde-Star), ZentralBlatt
Referenced by:: §35.10, 2nd item.
Encodings:: BibTeX

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T. H. Koornwinder and I. Sprinkhuizen-Kuyper (1978) Hypergeometric functions of

2\times 2

matrix argument are expressible in terms of Appel’s functions

F_{4}

. Proc. Amer. Math. Soc. 70 (1), pp. 39–42.

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External Links:: FullText, MathReview (P. N. Rathie), ZentralBlatt
Referenced by:: §35.7(ii).
Encodings:: BibTeX

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27: Bibliography M

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W. Magnus, F. Oberhettinger, and R. P. Soni (1966) Formulas and Theorems for the Special Functions of Mathematical Physics. 3rd edition, Springer-Verlag, New York-Berlin.

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Notes:: Table errata: Math. Comp. v. 23 (1969), p. 471, v. 24 (1970), pp. 240 and 505, v. 25 (1971), no. 113, p. 201, v. 29 (1975), no. 130, p. 672, v. 30 (1976), no. 135, pp. 677–678, v. 32 (1978), no. 141, pp. 319–320, v. 36 (1981), no. 153, pp. 315–317, v. 41 (1983), no. 164, pp. 775–776, and v. 81 (2012), no. 280, p. 2251
External Links:: MathReview (Mary L. Boas), ZentralBlatt
Referenced by:: §10.15, §10.22(vi), §10.32(iv), §10.38, §10.43(vi), §10.60(iv), §10.9(iv), §11.5(iii), Ch.11, §12.12, §12.5(iv), Ch.12, §13.10(vi), §13.23(iv), §14.1, §14.11, §14.12(ii), §14.18(iv), §14.25, §14.3(iii), §14.5(v), §14.7(iv), Ch.14, (25.5.21), §25.5, (25.6.1), (25.6.2), §25.6(i), §8.1, Notations B, Notations I, Notations P, Notations P, Notations Q, Notations Q.
Encodings:: BibTeX

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N. W. McLachlan (1947) Theory and Application of Mathieu Functions. Clarendon Press, Oxford.

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Notes:: Corrected republication by Dover Publications, Inc., New York, 1964.
External Links:: MathReview (M. J. O. Strutt), ZentralBlatt
Referenced by:: §28.1, §28.2(i), §28.29(i), 1st item, 1st item, §28.33(ii), §28.4(vi), §28.4(vi), §28.5(i), §28.5(i), §28.9, Ch.28, Notations F, Notations G, Notations M, Notations N.
Encodings:: BibTeX

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J. Meixner and F. W. Schäfke (1954) Mathieusche Funktionen und Sphäroidfunktionen mit Anwendungen auf physikalische und technische Probleme. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band LXXI, Springer-Verlag, Berlin (German).

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External Links:: MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: Table 28.1.1, §28.10(i), §28.11, §28.12(i), §28.12(ii), §28.12(iii), §28.14, §28.15(i), §28.15(ii), §28.16, §28.17, §28.19, §28.2(ii), §28.2(iii), §28.2(iv), §28.2(v), §28.2(vi), §28.20(ii), §28.20(iii), §28.20(vi), §28.20(vii), §28.22(i), §28.22(ii), §28.23, §28.24, §28.25, §28.28(i), 2nd item, 3rd item, §28.33(ii), item (e), item (d), §28.4(i), §28.4(ii), §28.4(iii), §28.4(iv), §28.4(v), §28.4(vi), §28.4(vi), §28.5(i), §28.5(i), §28.6(i), §28.6(ii), §28.7, §28.7, §28.8(i), §28.8(ii), §28.9, Ch.28, §30.1, §30.10, §30.11(iii), §30.11(iii), §30.11(iv), §30.11(v), §30.11(vi), §30.11(vi), §30.13(i), §30.13(ii), §30.13(iii), §30.13(iv), §30.13(v), §30.13(v), §30.14(i), §30.14(ii), §30.14(iii), §30.14(iv), §30.14(v), §30.14(v), §30.16(i), §30.16(i), §30.2(i), §30.2(ii), §30.3(i), §30.3(ii), §30.3(iii), §30.3(iv), §30.4(i), §30.4(ii), §30.4(iv), §30.5, §30.5, §30.6, §30.6, §30.8(i), §30.8(ii), §30.9(i), §30.9(i), §30.9(ii), §30.9(ii), Ch.30, Notations P, Notations P, Notations Q, Notations Q.
Encodings:: BibTeX

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L. M. Milne-Thomson (1933) The Calculus of Finite Differences. Macmillan and Co. Ltd., London.

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Notes:: Reprinted in 1951 by Macmillan, and in 1981 by Chelsea Publishing Co., New York, and American Mathematical Society, AMS Chelsea Book Series, Providence, RI.
External Links:: ZentralBlatt
Referenced by:: §24.16(i), §24.17(i), §24.2(i), §24.2(ii), §24.4(i), §24.4(ii), §24.4(iii), §24.4(v), §26.1, §26.1, Notations B, Notations B.
Encodings:: BibTeX

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R. J. Muirhead (1978) Latent roots and matrix variates: A review of some asymptotic results. Ann. Statist. 6 (1), pp. 5–33.

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External Links:: FullText, MathReview (Yasuko Chikuse), ZentralBlatt
Referenced by:: §35.6(i), §35.6(iii).
Encodings:: BibTeX

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28: Bibliography G

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G. H. Golub and C. F. Van Loan (1996) Matrix Computations. 3rd edition, Johns Hopkins University Press, Baltimore, MD.

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External Links:: ISBN 0-8018-5413-X; 0-8018-5414-8, MathReview, ZentralBlatt
Referenced by:: §3.2(i), §3.2(i), §3.2(ii), §3.2(vii).
Encodings:: BibTeX

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H. W. Gould (1960) Stirling number representation problems. Proc. Amer. Math. Soc. 11 (3), pp. 447–451.

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External Links:: FullText, MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §26.1, §26.1, Notations S, Notations S.
Encodings:: BibTeX

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C. H. Greene, U. Fano, and G. Strinati (1979) General form of the quantum-defect theory. Phys. Rev. A 19 (4), pp. 1485–1509.

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External Links:: Document
Referenced by:: item Greene et al. (1979):, Notations F, Notations F, Notations G.
Encodings:: BibTeX

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K. I. Gross and D. St. P. Richards (1987) Special functions of matrix argument. I. Algebraic induction, zonal polynomials, and hypergeometric functions. Trans. Amer. Math. Soc. 301 (2), pp. 781–811.

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External Links:: ISSN 0002-9947, FullText, MathReview (Wayne J. Holman, III), ZentralBlatt
Referenced by:: §35.8(i), §35.8(ii), §35.8(iv), Ch.35.
Encodings:: BibTeX

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K. I. Gross and D. St. P. Richards (1991) Hypergeometric functions on complex matrix space. Bull. Amer. Math. Soc. (N.S.) 24 (2), pp. 349–355.

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External Links:: ISSN 0273-0979, Document, MathReview, ZentralBlatt
Referenced by:: §35.7(iii), §35.8(i), §35.8(ii), §35.8(iii), §35.8(iv).
Encodings:: BibTeX

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29: 3.11 Approximation Techniques

… ►This is because in the notation of §3.11(i) … ►

3.11.26

F(s)=\mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(s\right)=\int_{0}^{\infty}e^{-% st}f(t)\,\mathrm{d}t

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Defines:: $F(s)$ : Laplace transform of $f(t)$ (locally)
Symbols:: $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Laplace transform, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm and $\int$ : integral
Keywords:: Laplace transform
A&S Ref:: 29.1.1
Permalink:: http://dlmf.nist.gov/3.11.E26
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.15):: The notation for the Laplace transform was changed to $\mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(z\right)$ from $\mathscr{L}(f;z)$ .
See also:: Annotations for §3.11(iv), §3.11(iv), §3.11 and Ch.3

… ►The matrix is symmetric and positive definite, but the system is ill-conditioned when

n

is large because the lower rows of the matrix are approximately proportional to one another. … ►Since

X_{k\ell}=X_{\ell k}

, the matrix is again symmetric. … ►The method of the fast Fourier transform (FFT) exploits the structure of the matrix

\boldsymbol{{\Omega}}

with elements

\omega_{n}^{jk}

j,k=0,1,\dots,n-1

. …

30: 28.2 Definitions and Basic Properties

… ►

28.2.3

(1-\zeta^{2})w^{\prime\prime}-\zeta w^{\prime}+\left(a+2q-4q\zeta^{2}\right)w=0.

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Symbols:: $q=h^{2}$ : parameter, $a$ : parameter, $w(z)$ : Mathieu’s equation solution and $\zeta$ : change of variable
A&S Ref:: 20.1.7 (in slightly different notation)
Permalink:: http://dlmf.nist.gov/28.2.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §28.2(i), §28.2 and Ch.28

… ►iff

e^{\pi\mathrm{i}\nu}

is an eigenvalue of the matrix … ►

§28.2(v) Eigenvalues $a_{n}$ , $b_{n}$

… ►When

\widehat{\nu}=0

1

, the notation for the two sets of eigenvalues corresponding to each

\widehat{\nu}

is shown in Table 28.2.1, together with the boundary conditions of the associated eigenvalue problem. … ►Table 28.2.2 gives the notation for the eigenfunctions corresponding to the eigenvalues in Table 28.2.1. …