modified Mathieu functions

SearchAdvancedHelp

(0.009 seconds)

21—30 of 31 matching pages

21: Bibliography F

… ►

Generalized parabolic cylinder functions. Asymptotic Anal. 5 (6), pp. 517–531.

ⓘ

External Links:

ISSN 0921-7134, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§12.15.

Encodings:

BibTeX

… ►

Expansions of hypergeometric functions in hypergeometric functions. Math. Comp. 15 (76), pp. 390–395.

ⓘ

External Links:

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

Referenced by:

§16.10.

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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).

Encodings:

BibTeX

… ►

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.

ⓘ

External Links:

ISSN 0378-4371, Document, MathReview

Referenced by:

6th item.

Encodings:

BibTeX

… ►

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

ⓘ

Notes:

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

External Links:

ISSN 0001-0782, Document, Remark

Referenced by:

§8.28(ii).

Encodings:

BibTeX

…

22: Bibliography C

… ►

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

ⓘ

External Links:

ISSN 0098-3500, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(iv).

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

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

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

§10.46.

Encodings:

BibTeX

… ►

Algorithm 597: Sequence of modified Bessel functions of the first kind. ACM Trans. Math. Software 9 (2), pp. 242–245.

ⓘ

Notes:

Double-precision Fortran, maximum accuracy 24S.

External Links:

ISSN 0098-3500, Document, GAMS (module 597; package TOMS), ZentralBlatt

Referenced by:

5th item.

Encodings:

BibTeX

… ►

Derivatives with respect to the degree and order of associated Legendre functions for $|z|>1$ using modified Bessel functions. Integral Transforms Spec. Funct. 21 (7-8), pp. 581–588.

ⓘ

External Links:

ISSN 1065-2469, Document, FullText, MathReview (Dmitriy Karp), ZentralBlatt

Referenced by:

§14.11, §14.11.

Encodings:

BibTeX

…

23: Bibliography M

… ►

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

ⓘ

External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

… ►

Theory and Application of Mathieu Functions. Clarendon Press, Oxford.

ⓘ

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

… ►

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

… ►

Mathieu Functions and Spheroidal Functions and Their Mathematical Foundations: Further Studies. Lecture Notes in Mathematics, Vol. 837, Springer-Verlag, Berlin-New York.

ⓘ

External Links:

ISBN 3-540-10282-5, 0-387-10282-5, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§28.10(ii), §28.11, §28.2(v), §28.27, §28.28(i), §28.28(v), §28.30(ii), §28.6(i), §28.6(i), §28.6(i), §28.7, §28.7, Ch.28, §30.11(vi), §30.12, §30.13(v), §30.14(v), §30.15(i), §30.3(iv), Ch.30, Profile Gerhard Wolf.

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

Document, ZentralBlatt

Referenced by:

§28.7.

Encodings:

BibTeX

…

24: Errata

… ►

Sentences were added just below (10.6.5) and (10.29.3) regarding results on modified quotients of the form $z{𝒞}_{\nu \pm 1}\left(z\right)/{𝒞}_{\nu }\left(z\right)$ and $z{𝒵}_{\nu \pm 1}\left(z\right)/{𝒵}_{\nu }\left(z\right)$, respectively (suggested by Art Ballato on 2021-04-29).

… ►

Bibliographic citations and clarifications have been added, removed, or modified in §§5.6(i), 5.10, 7.8, 7.25(iii), and 32.16.

… ►

Bibliographic citations, clarifications, typographical corrections and added or modified sentences appear.

… ►

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

… ►

Clarifications, typographic corrections, added or modified sentences appear in §§1.2(i), 1.10(i), 4.6(ii), 5.11(i), (11.11.1), (11.11.9), (21.5.7), and (27.14.7).

…

25: Bibliography T

… ►

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

ⓘ

Notes:

Algol program.

External Links:

Document, MathReview (Carl C. Grosjean), ZentralBlatt

Referenced by:

§10.74(i), §10.74(iv), §10.77(iii).

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

§10.74(viii).

Encodings:

BibTeX

… ►

The numerical computation of special functions by use of quadrature rules for saddle point integrals. II. Gamma functions, modified Bessel functions and parabolic cylinder functions. Report TW 183/78 Mathematisch Centrum, Amsterdam, Afdeling Toegepaste Wiskunde.

ⓘ

Notes:

Algol procedures, maximum accuracy 14S.

External Links:

FullText, ZentralBlatt

Referenced by:

§12.21(ii).

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

8th item.

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

§10.74(v).

Encodings:

BibTeX

…

26: Bibliography

… ►

Regular and irregular Coulomb wave functions expressed in terms of Bessel-Clifford functions. J. Math. Physics 33, pp. 111–116.

ⓘ

External Links:

MathReview (E. Weber), ZentralBlatt

Referenced by:

§33.9(ii).

Encodings:

BibTeX

… ►

Algorithm 804: Subroutines for the computation of Mathieu functions of integer orders. ACM Trans. Math. Software 26 (3), pp. 408–414.

ⓘ

Notes:

Double-precision C++, minimum accuracy 14D

External Links:

ISSN 0098-3500, Document, GAMS (module 804; package TOMS)

Referenced by:

1st item, 1st item.

Encodings:

BibTeX

… ►

Scattering by singular potentials with a perturbation – Theoretical introduction to Mathieu functions. J. Mathematical Phys. 16, pp. 961–970.

ⓘ

Notes:

Erratum: same journal 17 (1975), no. 7, p. 1361

External Links:

Document, MathReview (K. Veselic), ZentralBlatt

Referenced by:

4th item.

Encodings:

BibTeX

… ►

Computation of modified Bessel functions and their ratios. Math. Comp. 28 (125), pp. 239–251.

ⓘ

External Links:

ISSN 0025-5718, FullText, MathReview (L. Lorch), ZentralBlatt

Referenced by:

§10.74(v).

Encodings:

BibTeX

… ►

Periodic Differential Equations. An Introduction to Mathieu, Lamé, and Allied Functions. International Series of Monographs in Pure and Applied Mathematics, Vol. 66, Pergamon Press, The Macmillan Co., New York.

ⓘ

External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§28.1, §28.1, §28.10(i), §28.11, §28.12(ii), §28.17, §28.2(ii), §28.2(iii), §28.2(iv), §28.28(i), §28.28(ii), §28.29(i), §28.32(i), §28.32(ii), §28.4(i), §28.5(i), Ch.28, §29.1, §29.11, §29.12(i), §29.12(ii), §29.12(iii), §29.14, §29.15(ii), §29.15(ii), §29.17(i), §29.18(iii), Ch.29, §30.1, §30.10, §30.11(i), §30.11(vi), §30.2(ii), §30.4(iii), §30.6, §30.8(i), §30.8(ii), Ch.30, §31.4, §31.5, §31.9(ii), Notations F, Notations G, Notations M, Notations N.

Encodings:

BibTeX

…

27: Bibliography Z

… ►

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

ⓘ

External Links:

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

Referenced by:

§24.16(iii).

Encodings:

BibTeX

… ►

Numerical analysis of Struve functions with applications to other special functions. Ann. Numer. Math. 2 (1-4), pp. 199–208.

ⓘ

External Links:

ISSN 1021-2655, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§11.7(ii).

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

§28.36(iii), §28.36(iii).

Encodings:

BibTeX

… ►

The summation of series of hyperbolic functions. SIAM J. Math. Anal. 10 (1), pp. 192–206.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§4.41.

Encodings:

BibTeX

… ►

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

ⓘ

Notes:

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

External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

…

28: Bibliography L

… ►

Bounds for modified Bessel functions. J. Comput. Appl. Math. 34 (3), pp. 263–267.

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview (N. Hayek Calil), ZentralBlatt

Referenced by:

§10.37.

Encodings:

BibTeX

… ►

The solutions of the Mathieu equation with a complex variable and at least one parameter large. Trans. Amer. Math. Soc. 36 (3), pp. 637–695.

ⓘ

External Links:

ISSN 0002-9947, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

… ►

New method to obtain small parameter power series expansions of Mathieu radial and angular functions. Math. Comp. 78 (265), pp. 255–274.

ⓘ

External Links:

ISSN 0025-5718, Document, FullText, MathReview (Junesang Choi), ZentralBlatt

Referenced by:

§28.24, §28.24.

Encodings:

BibTeX

… ►

Algorithm 537: Characteristic values of Mathieu’s differential equation. ACM Trans. Math. Software 5 (1), pp. 112–117.

ⓘ

External Links:

ISSN 0098-3500, Document, GAMS (module 537; package TOMS)

Referenced by:

2nd item, 2nd item.

Encodings:

BibTeX

… ►

An application of the finite element approximation method to find the complex zeros of the modified Bessel function $K_n(z)$ . Math. Comp. 33 (148), pp. 1299–1306.

ⓘ

External Links:

ISSN 0025-5718, FullText, MathReview (R. P. Boas, Jr.), ZentralBlatt

Referenced by:

§10.74(vi), 2nd item.

Encodings:

BibTeX

…

29: Bibliography G

… ►

Inequalities for modified Bessel functions and their integrals. J. Math. Anal. Appl. 420 (1), pp. 373–386.

ⓘ

External Links:

ISSN 0022-247X, Document, Link, MathReview (Nihat Yagmur), ZentralBlatt

Referenced by:

§10.37, §10.37.

Encodings:

BibTeX

… ►

On the computation of modified Bessel function ratios. Math. Comp. 32 (143), pp. 865–875.

ⓘ

External Links:

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

Referenced by:

§10.74(v).

Encodings:

BibTeX

… ►

Some integrals involving three modified Bessel functions. I. J. Math. Phys. 27 (3), pp. 682–687.

ⓘ

External Links:

ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§10.22(iv), §10.43(iv).

Encodings:

BibTeX

… ►

Mathieu functions. Trans. Camb. Philos. Soc. 23, pp. 303–336.

ⓘ

External Links:

ZentralBlatt

Referenced by:

§28.26(i), §28.26(i), §28.8(iii).

Encodings:

BibTeX

… ►

Mathieu functions, a visual approach. Amer. J. Phys. 71 (3), pp. 233–242.

ⓘ

External Links:

Document

Referenced by:

§28.33(ii).

Encodings:

BibTeX

…

30: Bibliography R

… ►

Tablitsy modifitsirovannykh funktsii Besselya $K_{\frac{1}{2}+i\beta }(x)$ . “Nauka”, Moscow (Russian).

ⓘ

Notes:

In Russian. Translated title: Tables of Modified Bessel Functions $K_{\frac{1}{2}+i\beta }(x)$

External Links:

MathReview, ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

… ►

Elliptic Functions, Theta Functions, and Riemann Surfaces. The Williams & Wilkins Co., Baltimore, MD.

ⓘ

External Links:

MathReview (H. H. Martens), ZentralBlatt

Referenced by:

§20.11(iv), §20.12(ii).

Encodings:

BibTeX

… ►

Mathieu functions of integral orders and real arguments. IEEE Trans. Microwave Theory Tech. 28 (3), pp. 276–277.

ⓘ

Notes:

Includes Fortran program

External Links:

FullText

Referenced by:

§28.36(ii), §28.36(iii).

Encodings:

BibTeX

►

Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function. Mathematics 9 (16).

ⓘ

External Links:

Link, ISSN 2227-7390, Document

Referenced by:

§8.15.

Encodings:

BibTeX

… ►

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.

ⓘ

External Links:

ISBN 3-7643-6287-1, MathReview Entry, ZentralBlatt

Referenced by:

5th item.

Encodings:

BibTeX

…