Dirac equation

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21: Bibliography B

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A program for computing the Fermi-Dirac functions. Comput. Phys. Comm. 21 (3), pp. 315–322.

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Notes:

Double-precision Fortran, maximum accuracy 10D.

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Document, Code

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2nd item.

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Integral equations and exact solutions for the fourth Painlevé equation. Proc. Roy. Soc. London Ser. A 437, pp. 1–24.

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ISSN 0962-8444, FullText, MathReview (Mehmet Can), ZentralBlatt

Referenced by:

§32.11(v), §32.2(iii).

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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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Vortices in Ginzburg-Landau Equations. In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), pp. 11–19.

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ISSN 1431-0643, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§17.17.

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Ordinary differential equations. Fourth edition, John Wiley & Sons, Inc., New York.

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External Links:

ISBN 0-471-86003-4, MathReview (Michal Čverčko)

Referenced by:

§1.13(viii), §1.13(viii), §1.18(iv).

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22: Bibliography S

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A Gaussian quadrature for the calculation of generalized Fermi-Dirac integrals. Comput. Phys. Comm. 66 (2-3), pp. 271–275.

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External Links:

ISSN 0010-4655, Document, MathReview, ZentralBlatt

Referenced by:

§25.18(i).

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On the evaluation of the Fermi-Dirac integrals. Astrophys. J. 376 (1, part 1), pp. 364–366.

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ISSN 0004-637X, Document, MathReview (De Hui Chen)

Referenced by:

§25.18(i).

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Characterization of Jacobian varieties in terms of soliton equations. Invent. Math. 83 (2), pp. 333–382.

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ISSN 0020-9910, Document, MathReview (Bert van Geemen), ZentralBlatt

Referenced by:

§21.9.

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Non-linear integral equations for Heun functions. Proc. Edinburgh Math. Soc. (2) 16, pp. 281–289.

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External Links:

Document, MathReview (J. D. DePree), ZentralBlatt

Referenced by:

§31.10(ii).

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The linear differential equation whose solutions are the products of solutions of two given differential equations. J. Math. Anal. Appl. 98 (1), pp. 130–147.

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ISSN 0022-247X, Document, FullText, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§1.13(v).

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23: 18.36 Miscellaneous Polynomials

… ►These are OP’s on the interval $(-1,1)$ with respect to an orthogonality measure obtained by adding constant multiples of “Dirac delta weights” at $-1$ and $1$ to the weight function for the Jacobi polynomials. … ►Classes of such polynomials have been found that generalize the classical OP’s in the sense that they satisfy second order matrix differential equations with coefficients independent of the degree. … ►

18.36.2 $$L_n^{(-k)}\left(x\right)={(-1)}^k\frac{(n-k)!}{n!}{x}^k{L}_{n-k}^{(k)}\left(x\right),$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $!$: factorial (as in $n!$), $k$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.36.E2

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Addition (effective with 1.2.0):

This equation was added.

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Annotations for §18.36(v), §18.36 and Ch.18

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18.36.7 $$T_k(y)\equiv -x{y}^{\prime \prime }+\frac{x-k}{x+k}((x+k+1)y^{\prime }-y)=(n-1)y.$$

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Symbols:

$\equiv$: equals by definition, $y$: real variable, $k$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.36(vi)

Permalink:

http://dlmf.nist.gov/18.36.E7

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Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.36(vi), §18.36(vi), §18.36 and Ch.18

… ►In §18.39(i) it is seen that the functions, $\sqrt{w(x)}{\widehat{H}}_{n+3}\left(x\right)$, are solutions of a Schrödinger equation with a rational potential energy; and, in spite of first appearances, the Sturm oscillation theorem, Simon (2005c, Theorem 3.3, p. 35), is satisfied. …

24: 9.11 Products

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§9.11(i) Differential Equation

… ►For an integral representation of the Dirac delta involving a product of two $\mathrm{Ai}$ functions see §1.17(ii). …

25: Bibliography P

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Sur les équations différentielles du second ordre à points critiques fixès. C.R. Acad. Sc. Paris 143, pp. 1111–1117.

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External Links:

ZentralBlatt

Referenced by:

§32.2(iv).

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Smoothing of the Stokes phenomenon for high-order differential equations. Proc. Roy. Soc. London Ser. A 436, pp. 165–186.

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ISSN 0962-8444, FullText, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§2.11(v).

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Evaluation of Fermi-Dirac Integral. In Nonlinear Numerical Methods and Rational Approximation (Wilrijk, 1987), A. Cuyt (Ed.), Mathematics and Its Applications, Vol. 43, pp. 435–444.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§25.18(i).

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Evaluation of the Fermi-Dirac integral of half-integer order. Zastos. Mat. 21 (2), pp. 289–301.

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External Links:

ISSN 0044-1899, MathReview (Alan M. Cohen), ZentralBlatt

Referenced by:

§25.18(i).

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Numerical calculation of the generalized Fermi-Dirac integrals. Comput. Phys. Comm. 55 (2), pp. 127–136.

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External Links:

Document

Referenced by:

§25.18(i).

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26: 14.18 Sums

… ►For a series representation of the Dirac delta in terms of products of Legendre polynomials see (1.17.22). …

27: Bibliography

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Asymptotics of solutions of the generalized sine-Gordon equation, the third Painlevé equation and the d’Alembert equation. Dokl. Akad. Nauk SSSR 280 (2), pp. 265–268 (Russian).

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Notes:

English translation: Soviet Math. Dokl. 31(1985), no. 1, pp. 45–47

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ISSN 0002-3264, MathReview (M. S. Agranovich), ZentralBlatt

Referenced by:

§32.11(iv).

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Nonlinear chains and Painlevé equations. Phys. D 73 (4), pp. 335–351.

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ISSN 0167-2789, Document, MathReview (F. Pempinelli), ZentralBlatt

Referenced by:

§32.2(v).

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Rational function approximations for Fermi-Dirac integrals. The Astrophysical Journal Supplement Series 84, pp. 101–108.

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Notes:

Double-precision Fortran, maximum precision 8D.

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ISSN 0067-0049, Document, Code

Referenced by:

5th item, 1st item.

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The Whittaker-Hill equation and the wave equation in paraboloidal co-ordinates. Proc. Roy. Soc. Edinburgh Sect. A 67, pp. 265–276.

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External Links:

MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

§28.31(i), §28.31(ii), §28.31(ii), §28.32(ii).

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Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:

ISBN 0-89871-412-5, MathReview (Michael Hanke), ZentralBlatt

Referenced by:

§3.7(i).

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