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21: 8.27 Approximations

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Luke (1969b, pp. 25, 40–41) gives Chebyshev-series expansions for $\Gamma\left(a,\omega z\right)$ (by specifying parameters) with $1\leq\omega<\infty$ , and $\gamma\left(a,\omega z\right)$ with $0\leq\omega\leq 1$ ; see also Temme (1994b, §3).

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22: 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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6	11	23	1255	40	37338
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23: 27.2 Functions

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Table 27.2.1: Primes.

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$n$	$p_{n}$	$p_{n+10}$	$p_{n+20}$	$p_{n+30}$	$p_{n+40}$	$p_{n+50}$	$p_{n+60}$	$p_{n+70}$	$p_{n+80}$	$p_{n+90}$
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Table 27.2.2: Functions related to division.

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$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
1	1	1	1	14	6	4	24	27	18	4	40	40	16	8	90
2	1	2	3	15	8	4	24	28	12	6	56	41	40	2	42
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24: Bibliography W

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T. Watanabe, M. Natori, and T. Oguni (Eds.) (1994) Mathematical Software for the P.C. and Work Stations – A Collection of Fortran 77 Programs. North-Holland Publishing Co., Amsterdam.

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Notes:: Translated from the 1989 Japanese original. With 1 IBM-PC floppy disk (5.25 inch) containing a large general purpose mathematical software collection written in Fortran, including some special function codes. Double precision.
External Links:: ISBN 0-444-82000-0, MathReview, ZentralBlatt
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.
Encodings:: BibTeX

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G. Weiss (1965) Harmonic Analysis. In Studies in Real and Complex Analysis, I. I. Hirschman (Ed.), Studies in Mathematics, Vol. 3, pp. 124–178.

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External Links:: MathReview, ZentralBlatt
Referenced by:: §1.15(iii), §1.15(iv), §1.15(v).
Encodings:: BibTeX

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E. J. Weniger (2007) Asymptotic Approximations to Truncation Errors of Series Representations for Special Functions. In Algorithms for Approximation, A. Iske and J. Levesley (Eds.), pp. 331–348.

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External Links:: ISBN 3-540-33283-9, Document, MathReview Entry, ZentralBlatt
Referenced by:: §2.10(i).
Encodings:: BibTeX

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E. P. Wigner (1959) Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. Pure and Applied Physics. Vol. 5, Academic Press, New York.

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Notes:: Expanded and improved ed. Translated from the German by J. J. Griffin.
External Links:: ISBN 0-12-750550-4, MathReview (A. S. Wightman), ZentralBlatt
Referenced by:: §34.12.
Encodings:: BibTeX

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J. Wimp (1964) A class of integral transforms. Proc. Edinburgh Math. Soc. (2) 14, pp. 33–40.

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External Links:: Document, MathReview (B. R. Bhonsle), ZentralBlatt
Referenced by:: §13.23(iv).
Encodings:: BibTeX

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25: Bibliography H

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P. I. Hadži (1969) Certain integrals that contain a probability function and degenerate hypergeometric functions. Bul. Akad. S̆tiince RSS Moldoven 1969 (2), pp. 40–47 (Russian).

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External Links:: MathReview (Wazir Hasan Abdi)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

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M. Hauss (1998) A Boole-type Formula involving Conjugate Euler Polynomials. In Charlemagne and his Heritage. 1200 Years of Civilization and Science in Europe, Vol. 2 (Aachen, 1995), P.L. Butzer, H. Th. Jongen, and W. Oberschelp (Eds.), pp. 361–375.

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External Links:: ISBN 2-503-50674-7, MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

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N. J. Hitchin (1995) Poncelet Polygons and the Painlevé Equations. In Geometry and Analysis (Bombay, 1992), Ramanan (Ed.), pp. 151–185.

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External Links:: MathReview (Henrik Pedersen), ZentralBlatt
Referenced by:: §32.9(iii).
Encodings:: BibTeX

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M. H. Hull and G. Breit (1959) Coulomb Wave Functions. In Handbuch der Physik, Bd. 41/1, S. Flügge (Ed.), pp. 408–465.

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External Links:: MathReview (G. Källén)
Referenced by:: §33.2(iii), §33.23(vii), §33.23(vii), §33.24, §33.5(ii), §33.7, Ch.33.
Encodings:: BibTeX

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26: 28.35 Tables

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Ince (1932) includes eigenvalues $a_{n}$ , $b_{n}$ , and Fourier coefficients for $n=0$ or $1(1)6$ , $q=0(1)10(2)20(4)40$ ; 7D. Also $\operatorname{ce}_{n}\left(x,q\right)$ , $\operatorname{se}_{n}\left(x,q\right)$ for $q=0(1)10$ , $x=1(1)90$ , corresponding to the eigenvalues in the tables; 5D. Notation: $a_{n}=\mathit{be}_{n}-2q$ , $b_{n}=\mathit{bo}_{n}-2q$ .

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Ince (1932) includes the first zero for $\operatorname{ce}_{n}$ , $\operatorname{se}_{n}$ for $n=2(1)5$ or $6$ , $q=0(1)10(2)40$ ; 4D. This reference also gives zeros of the first derivatives, together with expansions for small $q$ .

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27: 6.4 Analytic Continuation

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6.4.3

E_{1}\left(ze^{\pm\pi i}\right)=\operatorname{Ein}\left(-z\right)-\ln z-\gamma% \mp\pi i,

|\operatorname{ph}z|\leq\pi

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Symbols:: $\gamma$ : Euler’s constant, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{Ein}\left(\NVar{z}\right)$ : complementary exponential integral, $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase and $z$ : complex variable
Referenced by:: §6.4
Permalink:: http://dlmf.nist.gov/6.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §6.4 and Ch.6

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6.4.4

\operatorname{Ci}\left(ze^{\pm\pi i}\right)=\pm\pi i+\operatorname{Ci}\left(z% \right),

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{Ci}\left(\NVar{z}\right)$ : cosine integral, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and $z$ : complex variable
Referenced by:: §6.4
Permalink:: http://dlmf.nist.gov/6.4.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §6.4 and Ch.6

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6.4.5

\operatorname{Chi}\left(ze^{\pm\pi i}\right)=\pm\pi i+\operatorname{Chi}\left(% z\right),

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\operatorname{Chi}\left(\NVar{z}\right)$ : hyperbolic cosine integral, $\mathrm{i}$ : imaginary unit and $z$ : complex variable
Referenced by:: §6.4
Permalink:: http://dlmf.nist.gov/6.4.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §6.4 and Ch.6

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6.4.6

\mathrm{f}\left(ze^{\pm\pi i}\right)=\pi e^{\mp iz}-\mathrm{f}\left(z\right),

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for sine and cosine integrals and $z$ : complex variable
Referenced by:: §6.12(ii), §6.4
Permalink:: http://dlmf.nist.gov/6.4.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §6.4 and Ch.6

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6.4.7

\mathrm{g}\left(ze^{\pm\pi i}\right)=\mp\pi ie^{\mp iz}+\mathrm{g}\left(z% \right).

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\mathrm{g}\left(\NVar{z}\right)$ : auxiliary function for sine and cosine integrals and $z$ : complex variable
Referenced by:: §6.12(ii), §6.4
Permalink:: http://dlmf.nist.gov/6.4.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §6.4 and Ch.6

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28: 6.14 Integrals

… ►For collections of integrals, see Apelblat (1983, pp. 110–123), Bierens de Haan (1939, pp. 373–374, 409, 479, 571–572, 637, 664–673, 680–682, 685–697), Erdélyi et al. (1954a, vol. 1, pp. 40–42, 96–98, 177–178, 325), Geller and Ng (1969), Gradshteyn and Ryzhik (2000, §§5.2–5.3 and 6.2–6.27), Marichev (1983, pp. 182–184), Nielsen (1906b), Oberhettinger (1974, pp. 139–141), Oberhettinger (1990, pp. 53–55 and 158–160), Oberhettinger and Badii (1973, pp. 172–179), Prudnikov et al. (1986b, vol. 2, pp. 24–29 and 64–92), Prudnikov et al. (1992a, §§3.4–3.6), Prudnikov et al. (1992b, §§3.4–3.6), and Watrasiewicz (1967).

29: Bibliography J

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L. Jacobsen, W. B. Jones, and H. Waadeland (1986) Further results on the computation of incomplete gamma functions. In Analytic Theory of Continued Fractions, II (Pitlochry/Aviemore, 1985), W. J. Thron (Ed.), Lecture Notes in Math. 1199, pp. 67–89.

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External Links:: MathReview (Marietta J. Tretter), ZentralBlatt
Referenced by:: §8.25(iv).
Encodings:: BibTeX

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D. J. Jeffrey and N. Murdoch (2017) Stirling Numbers, Lambert W and the Gamma Function. In Mathematical Aspects of Computer and Information Sciences, J. Blömer, I. S. Kotsireas, T. Kutsia, and D. E. Simos (Eds.), Cham, pp. 275–279.

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External Links:: ISBN 978-3-319-72453-9, ZentralBlatt
Referenced by:: §4.13.
Encodings:: BibTeX

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X.-S. Jin and R. Wong (1998) Uniform asymptotic expansions for Meixner polynomials. Constr. Approx. 14 (1), pp. 113–150.

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External Links:: ISSN 0176-4276, Document, MathReview (Richard B. Paris), ZentralBlatt
Referenced by:: §18.24, §2.4(vi).
Encodings:: BibTeX

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30: Bibliography G

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F. G. Garvan and M. E. H. Ismail (Eds.) (2001) Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Developments in Mathematics, Vol. 4, Kluwer Academic Publishers, Dordrecht.

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External Links:: ISBN 1-4020-0101-0, MathReview Entry
Referenced by:: Profile Mourad E.H. Ismail.
Encodings:: BibTeX

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G. Gasper (1975) Formulas of the Dirichlet-Mehler Type. In Fractional Calculus and its Applications, B. Ross (Ed.), Lecture Notes in Math., Vol. 457, pp. 207–215.

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External Links:: MathReview (Mourad E. H. Ismail), ZentralBlatt
Referenced by:: §18.10(i).
Encodings:: BibTeX

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W. Gautschi (1984) Questions of Numerical Condition Related to Polynomials. In Studies in Numerical Analysis, G. H. Golub (Ed.), pp. 140–177.

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External Links:: MathReview Entry, ZentralBlatt
Referenced by:: §3.8(vi).
Encodings:: BibTeX

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W. Gautschi (1996) Orthogonal Polynomials: Applications and Computation. In Acta Numerica, 1996, A. Iserles (Ed.), Acta Numerica, Vol. 5, pp. 45–119.

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External Links:: MathReview (Sven Ehrich), ZentralBlatt
Referenced by:: §3.5(v).
Encodings:: BibTeX

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A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:: Includes Fortran program claiming 12S accuracy
External Links:: ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt
Referenced by:: 3rd item, §10.77(ix).
Encodings:: BibTeX

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