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11: Bibliography I

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The factorization method. Rev. Modern Phys. 23 (1), pp. 21–68.

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Referenced by:

§18.38(iii).

Encodings:

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Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math. 21 (1), pp. 359–375.

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External Links:

ISSN 0035-7596, MathReview (Joaquin Bustoz), ZentralBlatt

Referenced by:

§18.30(i).

Encodings:

BibTeX

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Bounds for the small real and purely imaginary zeros of Bessel and related functions. Methods Appl. Anal. 2 (1), pp. 1–21.

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External Links:

ISSN 1073-2772, FullText, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.21(v).

Encodings:

BibTeX

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More on electrostatic models for zeros of orthogonal polynomials. Numer. Funct. Anal. Optim. 21 (1-2), pp. 191–204.

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External Links:

ISSN 0163-0563, FullText, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§18.39(v).

Encodings:

BibTeX

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12: 26.6 Other Lattice Path Numbers

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Table 26.6.1: Delannoy numbers $D(m,n)$.

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$m$	$n$
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1	1	3	5	7	9	11	13	15	17	19	21
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10	1	21	221	1561	8361	36365	1 34245	4 33905	12 56465	33 17445	80 97453

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Table 26.6.2: Motzkin numbers $M(n)$.

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$n$	$M(n)$	$n$	$M(n)$	$n$	$M(n)$	$n$	$M(n)$	$n$	$M(n)$
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1	1	5	21	9	835	13	41835	17	23 56779
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Table 26.6.3: Narayana numbers $N(n,k)$.

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$n$	$k$
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7	0	1	21	105	175	105	21	1
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13: 23.9 Laurent and Other Power Series

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23.9.6 $$\mathrm{\wp }\left({\omega }_j+t\right)=e_j+(3{e_j}^2-5c_2)t^2+(10c_2{e}_j+21c_3)t^4+(7c_2{e_j}^2+21c_3{e}_j+5c_2^2)t^6+O\left(t^8\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $𝕃$: lattice, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $c_n$ and $t$: variable

Referenced by:

§23.9

Permalink:

http://dlmf.nist.gov/23.9.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §23.9 and Ch.23

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14: 5.22 Tables

… ►For early tables for both real and complex variables see Fletcher et al. (1962), Lebedev and Fedorova (1960), and Luke (1975, p. 21). …

15: 26.2 Basic Definitions

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Table 26.2.1: Partitions $p\left(n\right)$.

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$n$	$p\left(n\right)$	$n$	$p\left(n\right)$	$n$	$p\left(n\right)$
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4	5	21	792	38	26015
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16: 34.1 Special Notation

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17: 9.10 Integrals

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9.10.14 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-pt}\mathrm{Ai}\left(t\right)dt={\mathrm{e}}^{-p^3/3}\left(\frac{1}{3}-\frac{p{}_1{}^{}F_1^{}(\frac{1}{3};\frac{4}{3};\frac{1}{3}{p}^3)}{3^{4/3}\mathrm{\Gamma }\left(\frac{4}{3}\right)}+\frac{p^2{}_1{}^{}F_1^{}(\frac{2}{3};\frac{5}{3};\frac{1}{3}{p}^3)}{3^{5/3}\mathrm{\Gamma }\left(\frac{5}{3}\right)}\right),$$ $p\in ℂ$.

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_1{}^{}F_1^{}(a;b;z)$: $=M(a,b,z)$ notation for the Kummer confluent hypergeometric function, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $p$: parameter

Keywords:

Laplace transform

Source:

Gibbs (1973, Problem 72–21)

Permalink:

http://dlmf.nist.gov/9.10.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(v), §9.10 and Ch.9

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9.10.15 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-pt}\mathrm{Ai}\left(-t\right)dt=\frac{1}{3}{\mathrm{e}}^{p^3/3}\left(\frac{\mathrm{\Gamma }(\frac{1}{3},\frac{1}{3}{p}^3)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}+\frac{\mathrm{\Gamma }(\frac{2}{3},\frac{1}{3}{p}^3)}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}\right),$$ $\mathrm{\Re }p>0$,

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part and $p$: parameter

Keywords:

Laplace transform

Source:

Gibbs (1973, Problem 72–21)

Permalink:

http://dlmf.nist.gov/9.10.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(v), §9.10 and Ch.9

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9.10.16 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-pt}\mathrm{Bi}\left(-t\right)dt=\frac{1}{\sqrt{3}}{\mathrm{e}}^{p^3/3}\left(\frac{\mathrm{\Gamma }(\frac{2}{3},\frac{1}{3}{p}^3)}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}-\frac{\mathrm{\Gamma }(\frac{1}{3},\frac{1}{3}{p}^3)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\right),$$ $\mathrm{\Re }p>0$.

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part and $p$: parameter

Keywords:

Laplace transform

Source:

Gibbs (1973, Problem 72–21)

Permalink:

http://dlmf.nist.gov/9.10.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(v), §9.10 and Ch.9

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18: 9.19 Approximations

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Moshier (1989, §6.14) provides minimax rational approximations for calculating $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$. They are in terms of the variable $\zeta$, where $\zeta =\frac{2}{3}{x}^{3/2}$ when $x$ is positive, $\zeta =\frac{2}{3}{(-x)}^{3/2}$ when $x$ is negative, and $\zeta =0$ when $x=0$. The approximations apply when , that is, when or . The precision in the coefficients is 21S.

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19: Bibliography B

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A Fortran-90 based multiprecision system. ACM Trans. Math. Software 21 (4), pp. 379–387.

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External Links:

ISSN 0098-3500, Document, ZentralBlatt

Referenced by:

§4.48(iii).

Encodings:

BibTeX

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Automatic computation of zeros of Bessel functions and other special functions. SIAM J. Sci. Comput. 21 (4), pp. 1458–1464.

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External Links:

ISSN 1064-8275, Document, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.74(vi), §3.8(iii).

Encodings:

BibTeX

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Uniform asymptotic smoothing of Stokes’s discontinuities. Proc. Roy. Soc. London Ser. A 422, pp. 7–21.

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External Links:

ISSN 0080-4630, FullText, MathReview (André Hautot), ZentralBlatt

Referenced by:

§2.11(iv).

Encodings:

BibTeX

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Algorithm 21: Bessel function for a set of integer orders. Comm. ACM 3 (11), pp. 600.

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Notes:

Includes Algol code, minimum accuracy 5S. For certification see same journal 8(1965), p. 219.

External Links:

ISSN 0001-0782, Document

Referenced by:

§10.77(ii).

Encodings:

BibTeX

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Error estimates for the solution of linear systems. SIAM J. Sci. Comput. 21 (2), pp. 764–781.

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External Links:

ISSN 1095-7197, Document, MathReview (Dimitrios Noutsos), ZentralBlatt

Referenced by:

§3.2(iii).

Encodings:

BibTeX

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20: DLMF Project News

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