DLMF: Untitled Document

… ►

A. J. MacLeod (1996b) Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

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Notes:: Includes Fortran code, maximum accuracy 20S.
External Links:: ISSN 1017-1398, Document, MathReview, ZentralBlatt
Referenced by:: §6.13, 4th item, §6.21(ii).
Encodings:: BibTeX

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W. Magnus and S. Winkler (1966) Hill’s Equation. Interscience Tracts in Pure and Applied Mathematics, No. 20, Interscience Publishers John Wiley & Sons, New York-London-Sydney.

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Notes:: Reprinted by Dover Publications, Inc., New York, 1979.
External Links:: MathReview (H. Hochstadt), ZentralBlatt
Referenced by:: §1.3(iii), §28.29(i), §28.29(ii), §28.29(ii), §28.29(iii), §28.29(iii), §28.29(iii), §28.30(i), Ch.28, §29.3(i), §29.9.
Encodings:: BibTeX

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Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

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External Links:: ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt
Referenced by:: §10.74(iii).
Encodings:: BibTeX

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R. Metzler, J. Klafter, and J. Jortner (1999) Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems. Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.

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External Links:: FullText
Referenced by:: §8.24(i).
Encodings:: BibTeX

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D. S. Moak (1981) The

q

-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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External Links:: ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.27(v).
Encodings:: BibTeX

…

… ►

§25.6(i) Function Values

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25.6.3

\zeta\left(-n\right)=-\frac{B_{n+1}}{n+1},

n=1,2,3,\dots

.

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function and $n$ : nonnegative integer
Source:: Apostol (1976, (20), p. 266); with $n\neq 0$
A&S Ref:: 23.2.15 (in slightly different form)
Permalink:: http://dlmf.nist.gov/25.6.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §25.6(i), §25.6 and Ch.25

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25.6.6

\zeta\left(2k+1\right)=\frac{(-1)^{k+1}(2\pi)^{2k+1}}{2(2k+1)!}\int_{0}^{1}B_{% 2k+1}\left(t\right)\cot\left(\pi t\right)\,\mathrm{d}t,

k=1,2,3,\dots

.

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Symbols:: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cot\NVar{z}$ : cotangent function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral and $k$ : nonnegative integer
Source:: Nörlund (1924, (81*), p. 66); with $\nu=k$
A&S Ref:: 23.2.17
Permalink:: http://dlmf.nist.gov/25.6.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §25.6(i), §25.6 and Ch.25

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§25.6(ii) Derivative Values

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§25.6(iii) Recursion Formulas

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6.16.1

\sin x+\tfrac{1}{3}\sin\left(3x\right)+\tfrac{1}{5}\sin\left(5x\right)+\dots=% \begin{cases}\frac{1}{4}\pi,&0<x<\pi,\\ 0,&x=0,\\ -\frac{1}{4}\pi,&-\pi<x<0.\end{cases}

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : sine function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/6.16.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §6.16(i), §6.16 and Ch.6

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6.16.2

S_{n}(x)=\sum_{k=0}^{n-1}\frac{\sin\left((2k+1)x\right)}{2k+1}=\frac{1}{2}\int% _{0}^{x}\frac{\sin\left(2nt\right)}{\sin t}\,\mathrm{d}t=\tfrac{1}{2}% \operatorname{Si}\left(2nx\right)+R_{n}(x),

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Defines:: $S_{n}(x)$ : partial sum (locally)
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $\operatorname{Si}\left(\NVar{z}\right)$ : sine integral, $x$ : real variable, $n$ : nonnegative integer and $R_{n}(x)$ : remainder term
Permalink:: http://dlmf.nist.gov/6.16.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §6.16(i), §6.16 and Ch.6

… ►The first maximum of

\frac{1}{2}\operatorname{Si}\left(x\right)

for positive

x

occurs at

x=\pi

and equals

(1.1789\dots)\times\frac{1}{4}\pi

; compare Figure 6.3.2. … ►It occurs with Fourier-series expansions of all piecewise continuous functions. … …

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R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:: ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:: Document, MathReview (Matti Jutila), ZentralBlatt
Referenced by:: §25.18(i).
Encodings:: BibTeX

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A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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External Links:: ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt
Referenced by:: §32.11(iv).
Encodings:: BibTeX

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T. H. Koornwinder (2009) The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.

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External Links:: ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt
Referenced by:: §18.26(v), §18.26(v).
Encodings:: BibTeX

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C. Krattenthaler (1993) HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively

q

-binomial sums and basic hypergeometric series. Séminaire Lotharingien de Combinatoire 30, pp. 61–76.

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Notes:: Also published in J. Symbolic Comput. (1995), v. 20, no. 5–6, pp. 737–744.
External Links:: Code, Document, MathReview (G. Gasper), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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… ►See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25). … ► … ►

§10.73(iii) Kelvin Functions

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§10.73(iv) Bickley Functions

…

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§28.8(ii) Sips’ Expansions

… ►Also let

\xi=2\sqrt{h}\cos x

and

D_{m}\left(\xi\right)=e^{-\ifrac{\xi^{2}}{4}}\mathit{He}_{m}\left(\xi\right)

(§18.3). … ►

Barrett’s Expansions

… ►The approximants are elementary functions, Airy functions, Bessel functions, and parabolic cylinder functions; compare §2.8. … ►

… ► ►►►

	Open Source	With Book	Commercial
…
20 Theta Functions
…

►‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … ►In the list below we identify four main sources of software for computing special functions. … ►

Commercial Software.

Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

… ►The following are web-based software repositories with significant holdings in the area of special functions. …

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Subsection 19.11(i)

A sentence and unnumbered equation

R_{C}\left(\gamma-\delta,\gamma\right)=\frac{-1}{\sqrt{\delta}}\operatorname{% arctan}\left(\frac{\sqrt{\delta}\sin\theta\sin\phi\sin\psi}{\alpha^{2}-1-% \alpha^{2}\cos\theta\cos\phi\cos\psi}\right),

were added which indicate that care must be taken with the multivalued functions in (19.11.5). See (Cayley, 1961, pp. 103-106).

Suggested by Albert Groenenboom.

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Equation (23.12.2)

23.12.2

\zeta\left(z\right)=\frac{{\pi}^{2}}{4\omega_{1}^{2}}\left(\frac{z}{3}+\frac{2% \omega_{1}}{\pi}\cot\left(\frac{\pi z}{2\omega_{1}}\right)-8\left(z-\frac{% \omega_{1}}{\pi}\sin\left(\frac{\pi z}{\omega_{1}}\right)\right)q^{2}+O\left(q% ^{4}\right)\right)

Originally, the factor of 2 was missing from the denominator of the argument of the $\cot$ function.

Reported by Blagoje Oblak on 2019-05-27

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Equation (22.19.2)

22.19.2

\sin\left(\tfrac{1}{2}\theta(t)\right)=\sin\left(\frac{1}{2}\alpha\right)% \operatorname{sn}\left(t+K,\sin\left(\tfrac{1}{2}\alpha\right)\right)

Originally the first argument to the function $\operatorname{sn}$ was given incorrectly as $t$ . The correct argument is $t+K$ .

Reported 2014-03-05 by Svante Janson.

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Chapters 8, 20, 36

Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

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References

Bibliographic citations were added in §§1.13(v), 10.14, 10.21(ii), 18.15(v), 18.32, 30.16(iii), 32.13(ii), and as general references in Chapters 19, 20, 22, and 23.

…

… ►The elementary symmetric functions of the zeros are (with

a_{n}\not=0

) … ►Resolvent cubic is

z^{3}+12z^{2}+20z+9=0

with roots

\theta_{1}=-1

,

\theta_{2}=-\tfrac{1}{2}(11+\sqrt{85})

,

\theta_{3}=-\tfrac{1}{2}(11-\sqrt{85})

, and

\sqrt{-\theta_{1}}=1

,

\sqrt{-\theta_{2}}=\tfrac{1}{2}(\sqrt{17}+\sqrt{5})

,

\sqrt{-\theta_{3}}=\tfrac{1}{2}(\sqrt{17}-\sqrt{5})

. … ►

1.11.23

\sqrt[n]{R}\left(\cos\left(\frac{\alpha+2k\pi}{n}\right)+\mathrm{i}\sin\left(% \frac{\alpha+2k\pi}{n}\right)\right),

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $k$ : integer, $n$ : nonnegative integer and $R$ : radius
Permalink:: http://dlmf.nist.gov/1.11.E23
Encodings:: pMML, png, TeX
See also:: Annotations for §1.11(iv), §1.11 and Ch.1

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B_{-2}^{5}=\tfrac{1}{120}(6-10t-15t^{2}+20t^{3}-5t^{4}),

… ►

§3.4(ii) Analytic Functions

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3.4.19

\frac{1}{k!}=\frac{1}{2\pi r^{k}}\int_{0}^{2\pi}e^{r\cos\theta}\cos\left(r\sin% \theta-k\theta\right)\,\mathrm{d}\theta.

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\int$ : integral, $\sin\NVar{z}$ : sine function and $r$ : radius
Permalink:: http://dlmf.nist.gov/3.4.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §3.4(ii), §3.4(ii), §3.4 and Ch.3

… ► … ►

3.4.33

\nabla^{4}u_{0,0}=\frac{1}{h^{4}}\,(20u_{0,0}-8(u_{1,0}+u_{0,1}+u_{-1,0}+u_{0,% -1})+2(u_{1,1}+u_{1,-1}+u_{-1,1}+u_{-1,-1})+(u_{0,2}+u_{2,0}+u_{-2,0}+u_{0,-2}% ))+O\left(h^{2}\right),

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding and $u$ : function
A&S Ref:: 25.3.32
Permalink:: http://dlmf.nist.gov/3.4.E33
Encodings:: pMML, png, TeX
See also:: Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

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trigonometric%20functions

11—20 of 27 matching pages

11: Bibliography M

12: 25.6 Integer Arguments

§25.6(i) Function Values

§25.6(ii) Derivative Values

§25.6(iii) Recursion Formulas

13: 6.16 Mathematical Applications

14: Bibliography K

15: 10.73 Physical Applications

§10.73(iii) Kelvin Functions

§10.73(iv) Bickley Functions

16: 28.8 Asymptotic Expansions for Large $q$

§28.8(ii) Sips’ Expansions

Barrett’s Expansions

17: Software Index

18: Errata

19: 1.11 Zeros of Polynomials

20: 3.4 Differentiation

§3.4(ii) Analytic Functions

trigonometric%20functions

11—20 of 27 matching pages

11: Bibliography M

12: 25.6 Integer Arguments

§25.6(i) Function Values

§25.6(ii) Derivative Values

§25.6(iii) Recursion Formulas

13: 6.16 Mathematical Applications

14: Bibliography K

15: 10.73 Physical Applications

§10.73(iii) Kelvin Functions

§10.73(iv) Bickley Functions

16: 28.8 Asymptotic Expansions for Large q

§28.8(ii) Sips’ Expansions

Barrett’s Expansions

17: Software Index

18: Errata

19: 1.11 Zeros of Polynomials

20: 3.4 Differentiation

§3.4(ii) Analytic Functions

16: 28.8 Asymptotic Expansions for Large $q$