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21: 18.16 Zeros

… ►Let $j_{\alpha ,m}$ be the $m$th positive zero of the Bessel function $J_{\alpha }\left(x\right)$ (§10.21(i)). … ►Let ${\varphi }_m=j_{\alpha ,m}/\rho$. … ►For $m=1,2,\mathrm{\dots },n$, and with $j_{\alpha ,m}$ as in §18.16(ii), … ►In the notation of this reference $x_{n,m}=u_{a,m}$, $\mu =\sqrt{2n+1}$, and $\alpha ={\mu }^{-\frac{4}{3}}{a}_m$. … ►

§18.16(vii) Discriminants

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22: Bibliography T

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Asymptotic estimates of Stirling numbers. Stud. Appl. Math. 89 (3), pp. 233–243.

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External Links:

ISSN 0022-2526, FullText, MathReview (E. Rodney Canfield), ZentralBlatt

Referenced by:

§26.8(vii).

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High Speed Numerical Integration of Fermi Dirac Integrals. Master’s Thesis, Naval Postgraduate School, Monterey, CA.

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External Links:

FullText

Referenced by:

§25.21(vii).

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The Theory of Functions. 2nd edition, Oxford University Press, Oxford.

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External Links:

ZentralBlatt

Referenced by:

§1.10(ix), §1.10(v), §1.8(i), §1.8(ii), §1.8(iii), §1.9(vii).

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Numerical Linear Algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:

ISBN 0-89871-361-7, MathReview (Moody T. Chu), ZentralBlatt

Referenced by:

§3.2(iii), §3.2(vii).

Encodings:

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Rational Chebyshev approximation for the Fermi-Dirac integral $F_{-3/2}(x)$ . Solid–State Electronics 41 (5), pp. 771–773.

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External Links:

Document

Referenced by:

§25.21(vii).

Encodings:

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23: Bibliography G

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On high precision methods for computing integrals involving Bessel functions. Math. Comp. 33 (147), pp. 1049–1057.

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External Links:

ISSN 0025-5718, FullText, MathReview (K. Jetter), ZentralBlatt

Referenced by:

§10.74(vii).

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On the generalization of a method for computing Bessel function integrals. J. Comput. Appl. Math. 6 (2), pp. 167–168.

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External Links:

ISSN 0771-050X, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

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The triplets of helium. Philos. Trans. Roy. Soc. London Ser. A 228, pp. 151–196.

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External Links:

Document, ZentralBlatt (Möglich, Dr. F.)

Referenced by:

§34.3(vii).

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Matrix Computations. 3rd edition, Johns Hopkins University Press, Baltimore, MD.

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External Links:

ISBN 0-8018-5413-X; 0-8018-5414-8, MathReview, ZentralBlatt

Referenced by:

§3.2(i), §3.2(i), §3.2(ii), §3.2(vii).

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Constructing wavefunctions for nonlocal potentials. J. Chem. Phys. 52, pp. 6211–6217.

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External Links:

Document, MathReview (D. B. Fairlie)

Referenced by:

(9.12.22), (9.12.23), §9.12(vii), §9.17(iii).

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24: 12.10 Uniform Asymptotic Expansions for Large Parameter

… ►These cases are treated in §§12.10(vii)–12.10(viii). … ►Higher polynomials $u_s(t)$ can be calculated from the recurrence relation …and the $v_s(t)$ then follow from … ►

§12.10(vii) Negative $a$, . Expansions in Terms of Airy Functions

… ►The coefficients $A_s(\zeta )$ and $B_s(\zeta )$ are given by …

25: 1.10 Functions of a Complex Variable

… ►Let $f_1(z)$ be analytic in a domain $D_1$. … ►Then … ►

§1.10(vii) Inverse Functions

… ►Suppose that …Let $w_0=f(z_0)$. …

26: Bibliography

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Some orthogonal $q$-polynomials. Math. Nachr. 30, pp. 47–61.

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External Links:

ISSN 0025-584X, Document, MathReview (A. E. Danese), ZentralBlatt

Referenced by:

§18.27(vii).

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Solid State Physics. Holt, Rinehart and Winston, New York.

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External Links:

ISBN 0-03-083993-9

Referenced by:

§1.18(vii), §4.44, §8.22(ii).

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Orthogonal Polynomials and Special Functions. CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 21, Society for Industrial and Applied Mathematics, Philadelphia, PA.

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External Links:

ISBN 0-89871-018-9, MathReview (G. Gasper), ZentralBlatt

Referenced by:

§18.10(ii), §18.18(vii), Profile Richard A. Askey.

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Continuous $q$-Hermite Polynomials when $q>1$ . In $q$-series and Partitions (Minneapolis, MN, 1988), IMA Vol. Math. Appl., Vol. 18, pp. 151–158.

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External Links:

MathReview (Walter Van Assche), ZentralBlatt

Referenced by:

§18.28(vii).

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Singular Continuous Spectrum for a Class of Almost Periodic Jacobi Matrices. Bulletin of the American Mathematical Society 6 (1), pp. 81–85.

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External Links:

MathReview Entry

Referenced by:

§1.18(vii).

Encodings:

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27: 32.7 Bäcklund Transformations

… ►Let $w_j(z_j)=w(z_j;{\alpha }_j,{\beta }_j,{\gamma }_j,{\delta }_j)$, $j=0,1,2$, be solutions of ${\text{P}}_{\text{V}}$ with … ►satisfies ${\text{P}}_{\text{V}}$ with … ►

§32.7(vii) Sixth Painlevé Equation

►Let $w_j(z_j)=w_j(z_j;{\alpha }_j,{\beta }_j,{\gamma }_j,{\delta }_j)$, $j=0,1,2,3$, be solutions of ${\text{P}}_{\text{VI}}$ with … ►Also, …

28: Bibliography Q

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“Best possible” upper and lower bounds for the zeros of the Bessel function $J_{\nu }(x)$ . Trans. Amer. Math. Soc. 351 (7), pp. 2833–2859.

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External Links:

ISSN 0002-9947, FullText, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.21(vii).

Encodings:

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29: 10.75 Tables

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Wills et al. (1982) tabulates $j_{0,m}$, $j_{1,m}$, $y_{0,m}$, $y_{1,m}$ for $m=1(1)30$, 35D.

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MacDonald (1989) tabulates the first 30 zeros, in ascending order of absolute value in the fourth quadrant, of the function $J_0\left(z\right)-\mathrm{i}{J}_1\left(z\right)$, 6D. (Other zeros of this function can be obtained by reflection in the imaginary axis).

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Abramowitz and Stegun (1964, Chapter 11) tabulates ${\int }_0^x{J}_0\left(t\right)dt$, ${\int }_0^x{Y}_0\left(t\right)dt$, $x=0(.1)10$, 10D; ${\int }_0^x{t}^{-1}(1-J_0\left(t\right))dt$, ${\int }_x^{\mathrm{\infty }}{t}^{-1}{Y}_0\left(t\right)dt$, $x=0(.1)5$, 8D.

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§10.75(vii) Integrals of Modified Bessel Functions

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Abramowitz and Stegun (1964, Chapter 11) tabulates ${\mathrm{e}}^{-x}{\int }_0^x{I}_0\left(t\right)dt$, ${\mathrm{e}}^x{\int }_x^{\mathrm{\infty }}{K}_0\left(t\right)dt$, $x=0(.1)10$, 7D; ${\mathrm{e}}^{-x}{\int }_0^x{t}^{-1}(I_0\left(t\right)-1)dt$, $x{\mathrm{e}}^x{\int }_x^{\mathrm{\infty }}{t}^{-1}{K}_0\left(t\right)dt$, $x=0(.1)5$, 6D.

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30: Bibliography N

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Error bounds for the large-argument asymptotic expansions of the Lommel and allied functions. Stud. Appl. Math. 140 (4), pp. 508–541.

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External Links:

ISSN 0022-2526, Document, Link, MathReview (Pedro J. Pagola), ZentralBlatt

Referenced by:

§11.11(i), §11.11(i).

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Elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 99–102.

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Notes:

Includes Algol program.

External Links:

MathReview

Referenced by:

§19.39(iii).

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On the calculation of elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 91–94.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§19.36(ii).

Encodings:

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COULN, a program for evaluating negative energy Coulomb functions. Comput. Phys. Comm. 33 (4), pp. 413–419.

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Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

2nd item, 7th item.

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The asymptotic behavior of the general real solution of the third Painlevé equation. Dokl. Akad. Nauk SSSR 283 (5), pp. 1161–1165 (Russian).

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Notes:

In Russian. English translation: Sov. Math., Dokl. 30(1988), no. 8, pp. 666–668.

External Links:

ISSN 0002-3264, MathReview (Vojislav Marić), ZentralBlatt

Referenced by:

§32.11(iv).

Encodings:

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