modified Mathieu equation

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11: Bibliography K

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Asymptotic behavior of the solutions of the Painlevé equation of the first kind. Differ. Uravn. 24 (10), pp. 1684–1695 (Russian).

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Notes:

In Russian. English translation: Differential Equations 24(1988), no. 10, pp. 1107–1115 (1989)

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ISSN 0374-0641, MathReview, ZentralBlatt

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§32.11(i).

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Calculation of modified Bessel functions in a complex domain. Zh. Vychisl. Mat. i Mat. Fiz. 24 (5), pp. 650–664.

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 24(3), pp. 15–24

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ISSN 0044-4669, Document, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§10.42, §10.74(iv).

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Tables of values of the modified Mathieu functions. Math. Comp. 14, pp. 118–129.

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FullText, MathReview (L. J. Slater), ZentralBlatt

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5th item.

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Asymptotic solution of Maxwell’s equations near caustics. Izv. Vuz. Radiofiz. 7, pp. 1049–1056.

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§36.14(i).

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The Korteweg-de Vries Equation and Related Evolution Equations. In Nonlinear Wave Motion (Proc. AMS-SIAM Summer Sem., Clarkson Coll. Tech., Potsdam, N.Y., 1972), A. C. Newell (Ed.), Lectures in Appl. Math., Vol. 15, pp. 61–83.

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MathReview (G. L. Lamb, Jr.), ZentralBlatt

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§22.19(iii).

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12: Errata

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28.28.21 $$\frac{4}{\pi }{\int }_0^{\pi /2}{𝒞}_{2\mathrm{\ell }+1}^{(j)}(2hR)\cos \left((2\mathrm{\ell }+1)\varphi \right){\mathrm{ce}}_{2m+1}(t,h^2)dt={(-1)}^{\mathrm{\ell }+m}{A}_{2\mathrm{\ell }+1}^{2m+1}(h^2){\mathrm{Mc}}_{2m+1}^{(j)}(z,h)$$

28.28.22 $$\frac{4}{\pi }{\int }_0^{\pi /2}{𝒞}_{2\mathrm{\ell }+1}^{(j)}(2hR)\sin \left((2\mathrm{\ell }+1)\varphi \right){\mathrm{se}}_{2m+1}(t,h^2)dt={(-1)}^{\mathrm{\ell }+m}{B}_{2\mathrm{\ell }+1}^{2m+1}(h^2){\mathrm{Ms}}_{2m+1}^{(j)}(z,h),$$

Originally the prefactor $\frac{4}{\pi }$ and upper limit of integration $\pi /2$ in these two equations were given incorrectly as $\frac{2}{\pi }$ and $\pi$.

Reported 2015-05-20 by Ruslan Kabasayev

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13: Bibliography F

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The Lamé wave equation. Uspekhi Mat. Nauk 44 (1(265)), pp. 123–144, 248 (Russian).

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Notes:

In Russian. English translation: Russian Math. Surveys, 44(1989), no. 1, pp. 153–180

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ISSN 0042-1316, Document, MathReview (R. G. Langebartel), ZentralBlatt

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§29.11.

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Finite Differences and Difference Equations in the Real Domain. Clarendon Press, Oxford.

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MathReview (W. J. Trjitzinsky), ZentralBlatt

Referenced by:

§26.1, §26.1, Notations S, Notations S.

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Algebraic methods to compute Mathieu functions. J. Phys. A 34 (17), pp. 3541–3551.

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ISSN 0305-4470, Document, Link, MathReview (Aleksander Denisiuk), ZentralBlatt

Referenced by:

§28.6(i), §28.6(i), §28.6(ii), §28.6(ii), §28.8(i), §28.8(i), §28.8(ii), §28.8(ii), Erratum (V1.1.10) for Sections 28.6(i), 28.6(ii), 28.8(i), 28.8(ii).

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Characteristic values of the integral equation satisfied by the Mathieu functions and its application to a system with chirality-pair interaction on a one-dimensional lattice. Phys. A 190 (3-4), pp. 346–362.

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ISSN 0378-4371, Document, MathReview

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6th item.

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Algorithm 435: Modified incomplete gamma function. Comm. ACM 15 (11), pp. 993–995.

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Notes:

Includes Fortran code. For a Remark see ACM Trans. Math. Software 4 (1978), no. 3, pp. 296–304.

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ISSN 0001-0782, Document, Remark

Referenced by:

§8.28(ii).

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14: Bibliography S

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Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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

Referenced by:

§28.8(iv).

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Uniform asymptotic expansions of modified Mathieu functions. J. Reine Angew. Math. 247, pp. 1–17.

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External Links:

MathReview (J. Meixner), ZentralBlatt

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§28.8(iv).

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The computation of eigenvalues and solutions of Mathieu’s differential equation for noninteger order. ACM Trans. Math. Software 19 (3), pp. 377–390.

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ISSN 0098-3500, Document, ZentralBlatt

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item (e).

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On the double points of a Mathieu equation. J. Comput. Appl. Math. 107 (1), pp. 111–125.

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ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§28.7.

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Représentation asymptotique de la solution générale de l’équation de Mathieu-Hill. Acad. Roy. Belg. Bull. Cl. Sci. (5) 51 (11), pp. 1415–1446.

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ZentralBlatt

Referenced by:

§28.8(ii).

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15: Bibliography C

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On Temme’s algorithm for the modified Bessel function of the third kind. ACM Trans. Math. Software 6 (4), pp. 581–586.

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ISSN 0098-3500, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(iv).

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Théorie Générale de L’Équation de Mathieu et de quelques autres Équations différentielles de la mécanique. Masson et Cie, Paris (French).

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MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§28.1, Notations C, Notations I, Notations I, Notations J, Notations J, Notations S.

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Elementary Differential Equations. Clarendon Press, Oxford.

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ISBN 0-19-853139-7, MathReview, ZentralBlatt

Referenced by:

§16.12.

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Incomplete Hankel and modified Bessel functions: A class of special functions for electromagnetics. IEEE Trans. Antennas and Propagation 52 (12), pp. 3373–3389.

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ISSN 0018-926X, Document, MathReview (T. Erber), ZentralBlatt

Referenced by:

§10.46.

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Algorithm 352: Characteristic values and associated solutions of Mathieu’s differential equation. Comm. ACM 12 (7), pp. 399–407.

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Notes:

Includes double-precision Fortran program, maximum accuracy 13D.

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ISSN 0001-0782, Document

Referenced by:

§28.36(ii), §28.36(iii).

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16: Bibliography T

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On the numerical evaluation of the modified Bessel function of the third kind. J. Comput. Phys. 19 (3), pp. 324–337.

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Notes:

Algol program.

External Links:

Document, MathReview (Carl C. Grosjean), ZentralBlatt

Referenced by:

§10.74(i), §10.74(iv), §10.77(iii).

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Steepest descent paths for integrals defining the modified Bessel functions of imaginary order. Methods Appl. Anal. 1 (1), pp. 14–24.

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ISSN 1073-2772, FullText, MathReview (Anatoly Kilbas), ZentralBlatt

Referenced by:

§10.74(viii).

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Eigenfunction Expansions Associated with Second Order Differential Equations, Part 2, Partial Differential Equations. Clarendon Press, Oxford.

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ISBN 0198533187, MathReview Entry

Referenced by:

§1.18(x).

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Special functions in phase space: Mathieu functions. J. Phys. A 31 (31), pp. 6725–6739.

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ISSN 0305-4470, Document, MathReview (Bing Hong Wang), ZentralBlatt

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8th item.

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Further comments on the computation of modified Bessel function ratios. Math. Comp. 35 (151), pp. 937–939.

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ISSN 0025-5718, FullText, Document, MathReview (James M. Horner), ZentralBlatt

Referenced by:

§10.74(v).

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17: Bibliography B

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Mathieu functions of general order: Connection formulae, base functions and asymptotic formulae. I–V. Philos. Trans. Roy. Soc. London Ser. A 301, pp. 75–162.

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ISSN 0080-4614, FullText, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§28.8(iv).

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Mathieu’s Equation for Complex Parameters. Tables of Characteristic Values. U.S. Government Printing Office, Washington, D.C..

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MathReview (John Todd), ZentralBlatt

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1st item.

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Table of characteristic values of Mathieu’s equation for large values of the parameter. J. Washington Acad. Sci. 45 (6), pp. 166–196.

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MathReview (L. Fox)

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3rd item.

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Numerical aspects of Mathieu eigenvalues. Rend. Circ. Mat. Palermo (2) 15, pp. 51–97.

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Document, MathReview (J. Meixner), ZentralBlatt

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item (e).

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A note on Mathieu functions. Proc. Nederl. Akad. Wetensch. 51 (7), pp. 891–893=Indagationes Math. 10, 319–321 (1948).

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MathReview (M. J. O. Strutt), ZentralBlatt

Referenced by:

§28.7.

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18: Bibliography M

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Chebyshev expansions for modified Struve and related functions. Math. Comp. 60 (202), pp. 735–747.

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ISSN 0025-5718, FullText, MathReview, ZentralBlatt

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2nd item.

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On reducing the Heun equation to the hypergeometric equation. J. Differential Equations 213 (1), pp. 171–203.

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ISSN 0022-0396, Document, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§31.7(i).

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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.

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

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

Referenced by:

§10.74(iii).

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The characteristic numbers of the Mathieu equation with purely imaginary parameter. Phil. Mag. Series 7 8 (53), pp. 834–840.

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Document, ZentralBlatt

Referenced by:

§28.7.

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19: Bibliography V

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Accurate calculation of the modified Mathieu functions of integer order. Quart. Appl. Math. 65 (1), pp. 1–23.

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Notes:

For software, see Van Buren (website)

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ISSN 0033-569X, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§28.36(iii).

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Mathieu and Spheroidal Wave Functions: Fortran Programs for their Accurate Calculation

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External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, A. L. Van Buren and J. E. Boisvert (2002), A. L. Van Buren and J. E. Boisvert (2004), A. L. Van Buren and J. E. Boisvert (2007).

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A Mathieu equation for ships rolling among waves. I, II. Norske Vid. Selsk. Forh., Trondheim 22 (25–26), pp. 113–123.

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MathReview (J. V. Wehausen), ZentralBlatt

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2nd item.

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On the growth of convergence radii for the eigenvalues of the Mathieu equation. Math. Nachr. 192, pp. 239–253.

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ISSN 0025-584X, Document, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§28.6(i).

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

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ISSN 0176-4276, Document, MathReview, ZentralBlatt

Referenced by:

§30.16(i), §30.16(ii), §30.16(ii).

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20: Bibliography Z

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On the Computation of Zeros of Bessel and Bessel-related Functions. In Proceedings of the Sixth International Colloquium on Differential Equations (Plovdiv, Bulgaria, 1995), D. Bainov (Ed.), Utrecht, pp. 409–416.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§10.74(vi).

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A modified Bernoulli number. Nieuw Arch. Wisk. (4) 16 (1-2), pp. 63–72.

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ISSN 0028-9825, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.16(iii).

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Error bounds for asymptotic solutions of second-order linear difference equations. J. Comput. Appl. Math. 71 (2), pp. 191–212.

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ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§2.9(i), §2.9(ii).

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Mathieu functions for purely imaginary parameters. J. Comput. Appl. Math. 236 (17), pp. 4513–4524.

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ISSN 0377-0427, Document, FullText, MathReview (Bartolomeu D. B. Figueiredo), ZentralBlatt

Referenced by:

§28.36(iii), §28.36(iii).

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Tablitsy modifitsirovannykh funktsii Besselya s mnimym indeksom $K_{i\tau }(x)$ . Vyčisl. Centr Akad. Nauk SSSR, Moscow.

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Notes:

Translated title: Tables of the Modified Bessel Functions with Imaginary Index $K_{i\tau }(x)$

External Links:

MathReview (John Todd), ZentralBlatt

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1st item.

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