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11: Bibliography S

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Ein Verfahren zur Berechnung des charakteristischen Exponenten der Mathieuschen Differentialgleichung III. Numer. Math. 8 (1), pp. 68–71.

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Notes:

Includes Algol program

External Links:

ISSN 0029-599X, Document, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§28.36(ii).

Encodings:

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Hypergeometric Functions and Their Applications. Texts in Applied Mathematics, Vol. 8, Springer-Verlag, New York.

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External Links:

ISBN 0-387-97558-6, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

Erratum (V1.2.0) Section 18.39.

Encodings:

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Coulomb functions analytic in the energy. Comput. Phys. Comm. 25 (1), pp. 87–95.

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Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

§33.1, §33.14(iv), 9th item.

Encodings:

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The Bound State of Weakly Coupled Schrödinger Operators in One and Two Dimensions. Annals of Physics 97 (2), pp. 279–288.

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External Links:

Document, MathReview Entry

Referenced by:

§1.18(viii).

Encodings:

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Resonances in $n$-body quantum systems with dilatation analytic potentials and the foundations of time-dependent perturbation theory. Ann. of Math. (2) 97, pp. 247–274.

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External Links:

ISSN 0003-486X, Document, Link, MathReview (K. Kumar)

Referenced by:

§1.18(viii).

Encodings:

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12: Bibliography L

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Some differential equations and associated integral equations. Quart. J. Math. (Oxford) 5, pp. 81–97.

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External Links:

Document, ZentralBlatt

Referenced by:

§31.10(i), §31.9(i).

Encodings:

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On the maxima and minima of Bernoulli polynomials. Amer. Math. Monthly 47 (8), pp. 533–538.

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External Links:

FullText, MathReview (W. E. Milne), ZentralBlatt

Referenced by:

§24.12(i), §24.9.

Encodings:

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A partition function connected with the modulus five. Duke Math. J. 8 (4), pp. 631–655.

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External Links:

Document, MathReview (R. D. James), ZentralBlatt

Referenced by:

§17.18.

Encodings:

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On the theory of diffraction by an aperture in an infinite plane screen. I. Phys. Rev. 74 (8), pp. 958–974.

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External Links:

Document, MathReview (C. J. Bouwkamp), ZentralBlatt

Referenced by:

§11.12.

Encodings:

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Solutions of the fifth Painlevé equation. Differ. Uravn. 4 (8), pp. 1413–1420 (Russian).

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Notes:

In Russian. English translation: Differential Equations 4(8), pp. 732–735

External Links:

MathReview (C. E. Langenhop), ZentralBlatt

Referenced by:

§32.10(v), §32.8(v).

Encodings:

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13: Bibliography E

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A formula including Legendre’s $E{K}^{\prime }+K{E}^{\prime }-K{K}^{\prime }=\frac{1}{2}\pi$ . Messenger of Math. 33, pp. 31–32.

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Notes:

Errata on p. 45 of the same volume.

External Links:

ZentralBlatt

Referenced by:

§15.16.

Encodings:

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Higher Transcendental Functions. Vol. II. McGraw-Hill Book Company, Inc., New York-Toronto-London.

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Notes:

Reprinted by Robert E. Krieger Publishing Co. Inc., 1981. Table errata: Math. Comp. v. 41 (1983), no. 164, p. 778, v. 36 (1981), no. 153, p. 315, v. 30 (1976), no. 135, p. 675, v. 29 (1975), no. 130, p. 670, v. 26 (1972), no. 118, p. 598, v. 25 (1971), no. 113, p. 199, v. 24 (1970), no. 109, p. 239, v. 17 (1963), no. 84, p. 485.

External Links:

MathReview (E. T. Copson), ZentralBlatt

Referenced by:

§10.22(iv), §10.22(vi), §10.23(iii), §10.23(iii), §10.23(iv), §10.32(i), §10.32(iii), §10.32(iv), §10.43(iv), §10.43(vi), §10.44(iv), §10.60(iv), §10.9(i), §10.9(iii), §10.9(iv), §11.10(vi), §11.4(i), Ch.11, §12.12, §12.12, §12.13(i), §12.5(iv), Ch.12, §18.16(vi), (18.17.21_1), §18.17(iv), §18.17(i), §18.17(viii), (18.35.1), (18.35.2), (18.35.4_5), §18.35(i), (18.5.11_1), (18.5.11_3), §18.5(ii), §18.9(iii), §19.1, §19.10(i), §19.14(ii), (19.25.40), §19.25(vi), §19.5, §19.7(ii), §19.9(i), Ch.19, §20.9(i), §22.15(ii), §23.6(iv), Ch.23, §8.13(i), §8.14, §8.3(i), §8.4, §8.6(iii), §8.7, Ch.8, Notations P, Notations P.

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Algorithm 934: Fortran 90 subroutines to compute Mathieu functions for complex values of the parameter. ACM Trans. Math. Softw. 40 (1), pp. 8:1–8:19.

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External Links:

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

Referenced by:

4th item, §28.36(iii).

Encodings:

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Algorithm 861: Fortran 90 subroutines for computing the expansion coefficients of Mathieu functions using Blanch’s algorithm. ACM Trans. Math. Software 32 (4), pp. 622–634.

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External Links:

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

Referenced by:

3rd item, §28.36(iii).

Encodings:

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A new series representation for $\zeta (3)$ . Amer. Math. Monthly 97 (3), pp. 219–220.

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External Links:

ISSN 0002-9890, FullText, MathReview (Christian Radoux), ZentralBlatt

Referenced by:

(25.8.10).

Encodings:

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14: Bibliography I

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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.

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Highly Oscillatory Quadrature: The Story So Far. In Numerical Mathematics and Advanced Applications, A. Bermudez de Castro and others (Eds.), pp. 97–118.

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Notes:

Proceedings of ENuMath, Santiago de Compostela (2005)

External Links:

ISBN 3-540-34287-7, MathReview, ZentralBlatt

Referenced by:

§3.5(vii).

Encodings:

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Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.

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External Links:

ISBN 978-0-521-78201-2; 0-521-78201-5, MathReview Entry, ZentralBlatt

Referenced by:

§1.4(v), Profile Mourad E.H. Ismail, Erratum (V1.0.20) for Chapter 18 Orthogonal Polynomials.

Encodings:

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Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.

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Notes:

With two chapters by Walter Van Assche, With a foreword by Richard A. Askey, Corrected reprint of the 2005 original.

External Links:

ISBN 978-0-521-14347-9, MathReview Entry, ZentralBlatt

Referenced by:

§18.12, (18.15.4_5), (18.16.19), (18.16.20), (18.16.21), §18.17(ii), §18.18(vii), §18.19, (18.2.20), §18.2(vi), §18.2(xii), §18.2(xii), §18.2(x), §18.20(i), §18.20(ii), §18.21(ii), §18.22(i), §18.22(ii), §18.22(iii), §18.23, §18.25(i), §18.25(iii), §18.26(i), §18.26(iv), (18.27.4), §18.27(i), §18.27(ii), §18.27(iii), §18.27(v), §18.27(vi), §18.28(i), §18.28(ii), §18.28(iii), §18.28(v), §18.28(vi), §18.28(vii), §18.28(viii), §18.30(vi), §18.30(vii), §18.30(iii), §18.30(iv), §18.30, §18.30(vi), §18.34(i), §18.34(ii), §18.34(ii), §18.34(iii), (18.35.4), (18.35.5), (18.35.6), (18.35.6_2), (18.35.6_3), (18.35.6_4), (18.35.7), (18.35.8), §18.35(ii), §18.35(iii), §18.36(iii), §18.38(ii), §18.39(iv), §18.39(iv), §18.39(iv), (18.40.4), Profile Mourad E.H. Ismail, Erratum (V1.0.20) for Chapter 18 Orthogonal Polynomials.

Encodings:

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15: Bibliography C

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Bessel functions $I_{\nu }(x)$ and $K_{\nu }(x)$ of real order and complex argument. Comput. Phys. Comm. 24 (1), pp. 97–105.

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Notes:

Double-precision Fortran, maximum accuracy 16S. For erratum see same journal 25 (1982), p. 207.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

2nd item.

Encodings:

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Elliptic integrals of the first kind. SIAM J. Math. Anal. 8 (2), pp. 231–242.

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External Links:

Document, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§19.29(iii).

Encodings:

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A Unified Approach to Recurrence Algorithms. In Approximation and Computation (West Lafayette, IN, 1993), R. V. M. Zahar (Ed.), International Series of Computational Mathematics, Vol. 119, pp. 97–120.

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External Links:

ISBN 0-8176-3753-2, MathReview (R. M. M. Mattheij), ZentralBlatt

Referenced by:

§3.6(vii).

Encodings:

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Coulomb phase shift. American Journal of Physics 47 (8), pp. 683–684.

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External Links:

Document

Referenced by:

§5.20.

Encodings:

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A connection formula for the second Painlevé transcendent. Arch. Rational Mech. Anal. 103 (2), pp. 97–138.

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External Links:

ISSN 0945-8396, Document, MathReview (Reinhard Schäfke), ZentralBlatt

Referenced by:

§32.11(ii).

Encodings:

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16: Bibliography F

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Calculation of elliptic integrals of the third kind by means of Gauss’ transformation. Math. Comp. 19 (89), pp. 97–104.

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External Links:

FullText, MathReview (J. E. L. Peck), ZentralBlatt

Referenced by:

§19.36(ii).

Encodings:

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Numerical calculation of singular integrals related to Hankel transform. Comput. Math. Appl. 21 (2-3), pp. 87–94.

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Notes:

Fortran, minimum precision 6D.

External Links:

ISSN 0898-1221, Document, MathReview, ZentralBlatt

Referenced by:

§10.77(ix).

Encodings:

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Algorithm 309. Gamma function with arbitrary precision. Comm. ACM 10 (8), pp. 511–512.

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External Links:

ISSN 0001-0782, Document

Referenced by:

§5.24(ii).

Encodings:

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Numerical evaluation of the elliptic integral of the third kind. Math. Comp. 19 (91), pp. 494–496.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§19.36(iv).

Encodings:

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Travel time surface of a transverse cusp caustic produced by reflection of acoustical transients from a curved metal surface. J. Acoust. Soc. Amer. 95 (2), pp. 650–660.

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External Links:

Document

Referenced by:

§36.14(iv).

Encodings:

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17: Bibliography G

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A continued fraction algorithm for the computation of higher transcendental functions in the complex plane. Math. Comp. 21 (97), pp. 18–29.

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External Links:

FullText, MathReview (D. C. Gilles), ZentralBlatt

Referenced by:

§10.74(v), §3.10(ii).

Encodings:

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Positive sums of the classical orthogonal polynomials. SIAM J. Math. Anal. 8 (3), pp. 423–447.

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External Links:

ISSN 0036-1410, Document, Link, MathReview (W. A. Al-Salam)

Referenced by:

(18.14.25), (18.14.26), (18.14.27).

Encodings:

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Algorithm 236: Bessel functions of the first kind. Comm. ACM 7 (8), pp. 479–480.

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Notes:

For certification see same journal 8(1965), pp. 105–106, for remark see ACM Trans. Math. Software, 1(1975), pp. 282–284. Includes Algol program, maximum accuracy: 11D.

External Links:

ISSN 0001-0782, Document

Referenced by:

§10.77(iii).

Encodings:

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Algorithm 259: Legendre functions for arguments larger than one. Comm. ACM 8 (8), pp. 488–492.

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Notes:

Variable-precision Algol. For remark see ACM Trans. Math. Software 3(2) (1977), p. 204–205

External Links:

ISSN 0001-0782, Document

Referenced by:

§14.34(ii).

Encodings:

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A theorem on the numerators of the Bernoulli numbers. Amer. Math. Monthly 97 (2), pp. 136–138.

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External Links:

ISSN 0002-9890, FullText, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.10(iv), §24.16(iii).

Encodings:

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18: 10.32 Integral Representations

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10.32.19 $$K_{\mu }\left(z\right)K_{\nu }\left(z\right)=\frac{1}{8\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(t+\frac{1}{2}\mu +\frac{1}{2}\nu \right)\mathrm{\Gamma }\left(t+\frac{1}{2}\mu -\frac{1}{2}\nu \right)\mathrm{\Gamma }\left(t-\frac{1}{2}\mu +\frac{1}{2}\nu \right)\mathrm{\Gamma }\left(t-\frac{1}{2}\mu -\frac{1}{2}\nu \right)}{\mathrm{\Gamma }\left(2t\right)}{(\frac{1}{2}z)}^{-2t}dt,$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $z$: complex variable and $\nu$: complex parameter

Keywords:

Mellin transform

Referenced by:

§10.32(iii)

Permalink:

http://dlmf.nist.gov/10.32.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §10.32(iii), §10.32(iii), §10.32 and Ch.10

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19: Bibliography D

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The principal frequencies of vibrating systems with elliptic boundaries. Quart. J. Mech. Appl. Math. 8 (3), pp. 361–372.

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External Links:

Document, MathReview (M. A. Hyman), ZentralBlatt

Referenced by:

3rd item.

Encodings:

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Bernoulli Numbers. Bibliography (1713–1990). Queen’s Papers in Pure and Applied Mathematics, Vol. 87, Queen’s University, Kingston, ON.

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Notes:

A frequently updated version, with links to reviews, exists online

External Links:

Online version, MathReview, ZentralBlatt

Referenced by:

Ch.24.

Encodings:

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Inequalities for extreme zeros of some classical orthogonal and $q$-orthogonal polynomials. Math. Model. Nat. Phenom. 8 (1), pp. 48–59.

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External Links:

ISSN 0973-5348, Document, FullText, MathReview (Ana Foulquié Moreno), ZentralBlatt

Referenced by:

§18.16(ii), §18.16(iv), §18.16(ii), §18.16(iv), §18.27(iv), §18.27(iv), §18.27(v), §18.27(v).

Encodings:

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Lamé instantons. J. High Energy Phys. 2000 (1), pp. Paper 19, 8.

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Notes:

(electronic only, article id. JHEP01(2000)019, 8 pp.).

External Links:

ISSN 1029-8479, Document, MathReview, ZentralBlatt

Referenced by:

§22.11.

Encodings:

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Functions with given moments and weight functions for orthogonal polynomials. Rocky Mountain J. Math. 23, pp. 87–104.

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External Links:

ISSN 0035-7596, MathReview (Olav Njåstad), ZentralBlatt

Referenced by:

§18.34(ii).

Encodings:

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20: Bibliography M

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Inequalities for the zeros of Bessel functions. SIAM J. Math. Anal. 8 (1), pp. 166–170.

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External Links:

ISSN 1095-7154, Document, MathReview (Mary L. Boas), ZentralBlatt

Referenced by:

§10.21(iv).

Encodings:

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On the $G$-function. VII, VIII. Nederl. Akad. Wetensch., Proc. 49, pp. 1063–1072, 1165–1175 = Indagationes Math. 8, 661–670, 713–723 (1946).

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§16.11(ii).

Encodings:

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Asymptotic analysis of edge-excited currents on a convex face of a perfectly conducting wedge under overlapping penumbra region conditions. IEEE Trans. Antennas and Propagation 44 (1), pp. 97–101.

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External Links:

Document

Referenced by:

§9.14.

Encodings:

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A class of generalized hypergeometric summations. J. Comput. Appl. Math. 87 (1), pp. 79–85.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§16.10.

Encodings:

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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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External Links:

Document, ZentralBlatt

Referenced by:

§28.7.

Encodings:

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