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11: Bibliography

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Algorithm 511: CDC 6600 subroutines IBESS and JBESS for Bessel functions $I_{\nu }(x)$ and $J_{\nu }(x)$, $x\ge 0$, $\nu \ge 0$ . ACM Trans. Math. Software 3 (1), pp. 93–95.

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Notes:

Single-precision Fortran, maximum accuracy 16S.

External Links:

ISSN 0098-3500, Document, Companion paper. Erratum. GAMS (module 511; package TOMS)

Referenced by:

1st item, 1st item.

Encodings:

BibTeX

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Arb: A C Library for Arbitrary Precision Ball Arithmetic.

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Notes:

An extended-precision C code for special functions. The library uses arbitrary-precision ball arithmetic. Ball arithmetic is an efficient technique for performing numerical computations with automatic and rigorous tracking of error bounds. Arb is designed with computer algebra and computational number theory in mind, extending the principles behind FLINT to the domain of real and complex numbers.

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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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).

Encodings:

BibTeX

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Center for Algorithms and Interactive Scientific Software.

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Notes:

Symbolic and extended-precision general purpose computer algebra system.

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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12: 31.15 Stieltjes Polynomials

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${\gamma }_j>0,$

… ►If the exponent and singularity parameters satisfy (31.15.5)–(31.15.6), then for every multi-index $𝐦=(m_1,m_2,\mathrm{\dots },m_{N-1})$, where each $m_j$ is a nonnegative integer, there is a unique Stieltjes polynomial with $m_j$ zeros in the open interval $(a_j,a_{j+1})$ for each $j=1,2,\mathrm{\dots },N-1$. …

13: Bibliography D

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Texas Instruments, Inc..

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Notes:

Symbolic and extended-precision numerical computing system.

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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An approximation for ${\int }_{\chi }^{\mathrm{\infty }}{e}^{-t^2/2}{t}^p𝑑t,$ $\chi >0,$ $p$ real. Math. Comp. 32 (141), pp. 271–275.

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External Links:

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

Referenced by:

1st item.

Encodings:

BibTeX

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14: 9.11 Products

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9.11.3 $${\mathrm{Ai}}^2\left(x\right)=\frac{1}{4\pi \sqrt{3}}{\int }_0^{\mathrm{\infty }}{J}_0\left(\frac{1}{12}{t}^3+xt\right)tdt,$$ $x\ge 0$,

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral and $x$: real variable

Source:

Lebedev (1965, Problem 22, p. 142)

Referenced by:

§9.5(i)

Permalink:

http://dlmf.nist.gov/9.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §9.11(iii), §9.11 and Ch.9

… ►For $\int z^n{w}_1{w}_{2}dz$, $\int z^n{w}_1{w}_2^{\prime }dz$, $\int z^n{w}_1^{\prime }{w}_2^{\prime }dz$, where $n$ is any positive integer, see Albright (1977). … ►

9.11.15 $${\int }_0^{\mathrm{\infty }}{t}^{\alpha -1}{\mathrm{Ai}}^2\left(t\right)dt=\frac{2\mathrm{\Gamma }\left(\alpha \right)}{{\pi }^{1/2}{12}^{(2\alpha +5)/6}\mathrm{\Gamma }\left(\frac{1}{3}\alpha +\frac{5}{6}\right)},$$ $\mathrm{\Re }\alpha >0$.

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral and $\mathrm{\Re }$: real part

Source:

Laurenzi (1993, p. 904)

Permalink:

http://dlmf.nist.gov/9.11.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §9.11(v), §9.11 and Ch.9

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15: 1.16 Distributions

… ►For $\alpha >-1$, … ►For $\alpha >0$, … ►where $n$ is an integer such that $\alpha +n>-1$. … ►where $k$ is a multi-index and $\left|k\right|=k_1+k_2+\mathrm{\cdots }+k_n$. … ►For a multi-index $𝜶=({\alpha }_1,{\alpha }_2,\mathrm{\dots },{\alpha }_n)$, define …