fundamental%20solutions

SearchAdvancedHelp

(0.003 seconds)

7 matching pages

1: Bibliography V

… ►

Integral representations for products of Lamé functions by use of fundamental solutions. SIAM J. Math. Anal. 15 (3), pp. 559–569.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.8.

Encodings:

BibTeX

… ►

Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.

ⓘ

External Links:

ISSN 0176-4276, Document, MathReview, ZentralBlatt

Referenced by:

§30.16(i), §30.16(ii), §30.16(ii).

Encodings:

BibTeX

… ►

On the rational solutions of the second Painlevé equation. Differ. Uravn. 1 (1), pp. 79–81 (Russian).

ⓘ

Notes:

In Russian. English translation: Differential Equations 1(1), pp. 58–59.

External Links:

MathReview, ZentralBlatt

Referenced by:

§32.8(ii).

Encodings:

BibTeX

…

2: 10.73 Physical Applications

… ►and on separation of variables we obtain solutions of the form ${\mathrm{e}}^{\pm \mathrm{i}n\varphi }{\mathrm{e}}^{\pm \kappa z}{J}_n\left(\kappa r\right)$, from which a solution satisfying prescribed boundary conditions may be constructed. … ►It is fundamental in the study of electromagnetic wave transmission. …See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25). … ►The functions ${𝗃}_n\left(x\right)$, ${𝗒}_n\left(x\right)$, ${𝗁}_n^{(1)}\left(x\right)$, and ${𝗁}_n^{(2)}\left(x\right)$ arise in the solution (again by separation of variables) of the Helmholtz equation in spherical coordinates $\rho ,\theta ,\varphi$ (§1.5(ii)): …In quantum mechanics the spherical Bessel functions arise in the solution of the Schrödinger wave equation for a particle in a central potential. …

3: Bibliography C

… ►

Analytical solution of a model for complex food webs. Phys. Rev. E 65 (3), pp. (030901–1)–(030901–4).

ⓘ

External Links:

Document

Referenced by:

§8.24(i).

Encodings:

BibTeX

… ►

Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

ⓘ

External Links:

ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt

Referenced by:

§26.8(vii).

Encodings:

BibTeX

… ►

Fourier, Gegenbauer and Jacobi expansions for a power-law fundamental solution of the polyharmonic equation and polyspherical addition theorems. SIGMA Symmetry Integrability Geom. Methods Appl. 9, pp. Paper 042, 26.

ⓘ

External Links:

ISSN 1815-0659, Document, MathReview (Heinrich Begehr), ZentralBlatt

Referenced by:

§14.28(ii), §14.28(ii).

Encodings:

BibTeX

… ►

Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

ⓘ

External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§13.29(v), §15.19(v).

Encodings:

BibTeX

… ►

Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.

ⓘ

External Links:

Document

Referenced by:

5th item.

Encodings:

BibTeX

…

4: Bibliography M

… ►

Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

ⓘ

Notes:

Includes Fortran code, maximum accuracy 20S.

External Links:

ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§6.13, 4th item, §6.21(ii).

Encodings:

BibTeX

… ►

Foundations of Finite Precision Rational Arithmetic. In Fundamentals of Numerical Computation (Computer-oriented Numerical Analysis), G. Alefeld and R. D. Grigorieff (Eds.), Comput. Suppl., Vol. 2, Vienna, pp. 85–111.

ⓘ

External Links:

MathReview (G. Blanch), ZentralBlatt

Referenced by:

§3.1(iii).

Encodings:

BibTeX

… ►

Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

ⓘ

External Links:

ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§10.74(iii).

Encodings:

BibTeX

… ►

The $q$-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

ⓘ

External Links:

ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.27(v).

Encodings:

BibTeX

… ►

CODATA recommended values of the fundamental physical constants: 2002. Rev. Mod.Phys. 77, pp. 1–107.

ⓘ

Referenced by:

§18.39(ii).

Encodings:

BibTeX

…

5: Bibliography K

… ►

Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

ⓘ

External Links:

ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

… ►

Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

ⓘ

External Links:

Document, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

… ►

Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

ⓘ

External Links:

ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt

Referenced by:

§32.11(iv).

Encodings:

BibTeX

… ►

The Art of Computer Programming. Vol. 1: Fundamental Algorithms. 1st edition, Addison-Wesley Publishing Co., Reading, MA-London-Don Mills, Ont.

ⓘ

Notes:

Second printing of second edition by Addison-Wesley Publishing Co., Reading, MA-London-Amsterdam, 1975.

External Links:

MathReview (M. Muller), ZentralBlatt

Referenced by:

§1.4(iii), (25.11.33).

Encodings:

BibTeX

… ►

The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.

ⓘ

External Links:

ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt

Referenced by:

§18.26(v), §18.26(v).

Encodings:

BibTeX

…

6: Bibliography G

… ►

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.

ⓘ

External Links:

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

Referenced by:

§35.2, §35.3(ii).

Encodings:

BibTeX

… ►

Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.

ⓘ

Notes:

For remark see same journal 24(1998), p. 355.

External Links:

Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt

Referenced by:

2nd item, §3.5(v), §3.5(v).

Encodings:

BibTeX

… ►

Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

ⓘ

Notes:

Includes Fortran program claiming 12S accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §10.77(ix).

Encodings:

BibTeX

… ►

Quantum mechanics: fundamentals. Second edition, Springer-Verlag, New York.

ⓘ

External Links:

ISBN 0-387-22023-2, MathReview (Howard E. Brandt), ZentralBlatt

Referenced by:

§18.39(i).

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

§18.38(iii).

Encodings:

BibTeX

…

7: 15.10 Hypergeometric Differential Equation

… ►

§15.10(i) Fundamental Solutions

… ►When none of the exponent pairs differ by an integer, that is, when none of $c$, $c-a-b$, $a-b$ is an integer, we have the following pairs $f_1(z)$, $f_2(z)$ of fundamental solutions. … ► … ►The three pairs of fundamental solutions given by (15.10.2), (15.10.4), and (15.10.6) can be transformed into 18 other solutions by means of (15.8.1), leading to a total of 24 solutions known as Kummer’s solutions. … ►The $\left(\genfrac{}{}{0.0pt}{}{6}{3}\right)=20$ connection formulas for the principal branches of Kummer’s solutions are: …