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1: Bibliography N

… ►

W. J. Nellis and B. C. Carlson (1966) Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

ⓘ

External Links:: FullText, MathReview (I. A. Stegun), ZentralBlatt
Referenced by:: §19.37(iv).
Encodings:: BibTeX

… ►

E. Neuman (1969b) On the calculation of elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 91–94.

ⓘ

External Links:: MathReview, ZentralBlatt
Referenced by:: §19.36(ii).
Encodings:: BibTeX

… ►

E. Neuman (2004) Inequalities involving Bessel functions of the first kind. JIPAM. J. Inequal. Pure Appl. Math. 5 (4), pp. Article 94, 4 pp. (electronic).

ⓘ

External Links:: ISSN 1443-5756, FullText, MathReview, ZentralBlatt
Referenced by:: §10.14.
Encodings:: BibTeX

… ►

N. E. Nørlund (1955) Hypergeometric functions. Acta Math. 94, pp. 289–349.

ⓘ

External Links:: Document, MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §16.10, §16.8(ii), §16.8(ii).
Encodings:: BibTeX

… ►

H. M. Nussenzveig (1965) High-frequency scattering by an impenetrable sphere. Ann. Physics 34 (1), pp. 23–95.

ⓘ

External Links:: Document, MathReview (E. Ar)
Referenced by:: §14.31(iii).
Encodings:: BibTeX

…

2: 15.7 Continued Fractions

… ►

15.7.1

\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a,b+1;c+1;z\right)}=t_{0% }-\cfrac{u_{1}z}{t_{1}-\cfrac{u_{2}z}{t_{2}-\cfrac{u_{3}z}{t_{3}-\cdots}}},

ⓘ

Symbols:: $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $t_{n}$ : coefficient and $u_{n}$ : coefficient
Permalink:: http://dlmf.nist.gov/15.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §15.7 and Ch.15

… ►

15.7.3

\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a,b+1;c+1;z\right)}=v_{0% }-\cfrac{w_{1}}{v_{1}-\cfrac{w_{2}}{v_{2}-\cfrac{w_{3}}{v_{3}-\cdots}}},

ⓘ

Symbols:: $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $v_{n}$ : coefficient and $w_{n}$ : coefficient
Permalink:: http://dlmf.nist.gov/15.7.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §15.7 and Ch.15

… ►

15.7.5

\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a+1,b+1;c+1;z\right)}={x% _{0}+\cfrac{y_{1}}{x_{1}+\cfrac{y_{2}}{x_{2}+\cfrac{y_{3}}{x_{3}+\cdots}}}},

ⓘ

Symbols:: $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $x$ : real variable, $y$ : real variable, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.7.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §15.7 and Ch.15

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3: 25.16 Mathematical Applications

… ►

25.16.2

\psi\left(x\right)=x-\frac{\zeta'\left(0\right)}{\zeta\left(0\right)}-\sum_{% \rho}\frac{x^{\rho}}{\rho}+o\left(1\right),

x\to\infty

ⓘ

Symbols:: $\psi\left(\NVar{x}\right)$ : Chebyshev $\psi$ -function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $o\left(\NVar{x}\right)$ : order less than and $x$ : real variable
Keywords:: asymptotic approximation
Source:: Apostol (2000, p. 9)
Permalink:: http://dlmf.nist.gov/25.16.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §25.16(i), §25.16 and Ch.25

… ►where

H_{n}

is given by (25.11.33). … ►

H\left(s\right)

has a simple pole with residue

\zeta\left(1-2r\right)

(

=-B_{2r}/(2r)

) at each odd negative integer

s=1-2r

r=1,2,3,\dots

. … ►

25.16.14

\sum_{r=1}^{\infty}\sum_{k=1}^{r}\frac{1}{rk(r+k)}=\frac{5}{4}\zeta\left(3% \right),

ⓘ

Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function and $k$ : nonnegative integer
Keywords:: infinite series, special value
Source:: Apostol and Vu (1984, (14), p. 94)
Permalink:: http://dlmf.nist.gov/25.16.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §25.16(ii), §25.16 and Ch.25

►

25.16.15

\sum_{r=1}^{\infty}\sum_{k=1}^{r}\frac{1}{r^{2}(r+k)}=\frac{3}{4}\zeta\left(3% \right).

ⓘ

Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function and $k$ : nonnegative integer
Keywords:: infinite series, special value
Source:: Apostol and Vu (1984, (15), p. 94)
Permalink:: http://dlmf.nist.gov/25.16.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §25.16(ii), §25.16 and Ch.25

…

4: Bibliography Z

… ►

M. R. Zaghloul (2016) Remark on “Algorithm 916: computing the Faddeyeva and Voigt functions”: efficiency improvements and Fortran translation. ACM Trans. Math. Softw. 42 (3), pp. 26:1–26:9.

ⓘ

Notes:: Includes MATLAB and Fortran programs claiming 6S or 13S accuracy for single and double precision
External Links:: ISSN 0098-3500, Link, Document, MathReview Entry
Referenced by:: 3rd item, 6th item, §7.25(iii), §7.25(vi).
Encodings:: BibTeX

►

M. R. Zaghloul (2017) Algorithm 985: Simple, Efficient, and Relatively Accurate Approximation for the Evaluation of the Faddeyeva Function. ACM Trans. Math. Softw. 44 (2), pp. 22:1–22:9.

ⓘ

External Links:: ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt
Referenced by:: 4th item, §7.25(iii).
Encodings:: BibTeX

… ►

J. Zhang (1996) A note on the

\tau

-method approximations for the Bessel functions

Y_{0}(z)

and

Y_{1}(z)

. Comput. Math. Appl. 31 (9), pp. 63–70.

ⓘ

External Links:: ISSN 0898-1221, Document, MathReview, ZentralBlatt
Referenced by:: §10.76(ii).
Encodings:: BibTeX

►

J. Zhang and J. A. Belward (1997) Chebyshev series approximations for the Bessel function

Y_{n}(z)

of complex argument. Appl. Math. Comput. 88 (2-3), pp. 275–286.

ⓘ

External Links:: ISSN 0096-3003, Document, MathReview (Sven Ehrich), ZentralBlatt
Referenced by:: §10.76(ii).
Encodings:: BibTeX

… ►

A. Zhedanov (1998) On some classes of polynomials orthogonal on arcs of the unit circle connected with symmetric orthogonal polynomials on an interval. J. Approx. Theory 94 (1), pp. 73–106.

ⓘ

External Links:: ISSN 0021-9045, Link, MathReview (T. S. Chihara), ZentralBlatt
Referenced by:: §18.33(iii).
Encodings:: BibTeX

…

H. E. Fettis and J. C. Caslin (1964) Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

ⓘ

Notes:: Reviewed in Math. Comp. v. 1919(1965)509. Table erratum: Math. Comp. v. 20 (1966), no. 96, pp. 639-640.
Referenced by:: §19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.37(iii).
Encodings:: BibTeX

… ►

F. Feuillebois (1991) Numerical calculation of singular integrals related to Hankel transform. Comput. Math. Appl. 21 (2-3), pp. 87–94.

ⓘ

Notes:: Fortran, minimum precision 6D.
External Links:: ISSN 0898-1221, Document, MathReview, ZentralBlatt
Referenced by:: §10.77(ix).
Encodings:: BibTeX

… ►

A. M. S. Filho and G. Schwachheim (1967) Algorithm 309. Gamma function with arbitrary precision. Comm. ACM 10 (8), pp. 511–512.

ⓘ

External Links:: ISSN 0001-0782, Document
Referenced by:: §5.24(ii).
Encodings:: BibTeX

… ►

S. R. Finch (2003) Mathematical Constants. Encyclopedia of Mathematics and its Applications, Vol. 94, Cambridge University Press, Cambridge.

ⓘ

External Links:: ISBN 0-521-81805-2, MathReview (E. J. Barbeau), ZentralBlatt
Referenced by:: §3.12.
Encodings:: BibTeX

… ►

C. K. Frederickson and P. L. Marston (1994) Travel time surface of a transverse cusp caustic produced by reflection of acoustical transients from a curved metal surface. J. Acoust. Soc. Amer. 95 (2), pp. 650–660.

ⓘ

External Links:: Document
Referenced by:: §36.14(iv).
Encodings:: BibTeX

…

6: Bibliography W

… ►

X. Wang and A. K. Rathie (2013) Extension of a quadratic transformation due to Whipple with an application. Adv. Difference Equ., pp. 2013:157, 8.

ⓘ

External Links:: ISSN 1687-1847, Document, FullText, MathReview (Daniele Ritelli), ZentralBlatt
Referenced by:: §16.6, §16.6.
Encodings:: BibTeX

… ►

Z. Wang and R. Wong (2003) Asymptotic expansions for second-order linear difference equations with a turning point. Numer. Math. 94 (1), pp. 147–194.

ⓘ

External Links:: ISSN 0029-599X, Document, MathReview (Sergey M. Zagorodnyuk), ZentralBlatt
Referenced by:: §2.9(iii).
Encodings:: BibTeX

… ►

J. K. G. Watson (1999) Asymptotic approximations for certain

6

j

and

9

j

symbols. J. Phys. A 32 (39), pp. 6901–6902.

ⓘ

External Links:: ISSN 0305-4470, Document, MathReview, ZentralBlatt
Referenced by:: §34.8, §34.8.
Encodings:: BibTeX

… ►

R. Wong and H. Li (1992b) Asymptotic expansions for second-order linear difference equations. J. Comput. Appl. Math. 41 (1-2), pp. 65–94.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview (Shao Zhu Chen), ZentralBlatt
Referenced by:: §2.9(i), §2.9(ii).
Encodings:: BibTeX

… ►

P. Wynn (1966) Upon systems of recursions which obtain among the quotients of the Padé table. Numer. Math. 8 (3), pp. 264–269.

ⓘ

External Links:: ISSN 0029-599X, Document, MathReview (F. L. Bauer), ZentralBlatt
Referenced by:: §3.11(iv).
Encodings:: BibTeX

7: DLMF Project News

error generating summary

8: 4.26 Integrals

… ►Extensive compendia of indefinite and definite integrals of trigonometric and inverse trigonometric functions include Apelblat (1983, pp. 48–109), Bierens de Haan (1939), Gradshteyn and Ryzhik (2000, Chapters 2–4), Gröbner and Hofreiter (1949, pp. 116–139), Gröbner and Hofreiter (1950, pp. 94–160), and Prudnikov et al. (1986a, §§1.5, 1.7, 2.5, 2.7).

9: Bibliography

… ►

M. Abramowitz and P. Rabinowitz (1954) Evaluation of Coulomb wave functions along the transition line. Physical Rev. (2) 96, pp. 77–79.

ⓘ

External Links:: Document, MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §33.12(i).
Encodings:: BibTeX

… ►

G. E. Andrews and R. Askey (1985) Classical Orthogonal Polynomials. In Orthogonal Polynomials and Applications, C. Brezinski, A. Draux, A. P. Magnus, P. Maroni, and A. Ronveaux (Eds.), Lecture Notes in Math., Vol. 1171, pp. 36–62.

ⓘ

External Links:: MathReview, ZentralBlatt
Referenced by:: (18.27.12_5), (18.27.6).
Encodings:: BibTeX

►

G. E. Andrews and D. Foata (1980) Congruences for the

q

-secant numbers. European J. Combin. 1 (4), pp. 283–287.

ⓘ

External Links:: ISSN 0195-6698, MathReview (J. E. Cigler), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

►

G. E. Andrews, I. P. Goulden, and D. M. Jackson (1986) Shanks’ convergence acceleration transform, Padé approximants and partitions. J. Combin. Theory Ser. A 43 (1), pp. 70–84.

ⓘ

External Links:: ISSN 0097-3165, Document, MathReview (Kenneth A. Jukes), ZentralBlatt
Referenced by:: §17.18.
Encodings:: BibTeX

… ►

T. M. Apostol (1985b) Note on the trivial zeros of Dirichlet

L

-functions. Proc. Amer. Math. Soc. 94 (1), pp. 29–30.

ⓘ

External Links:: ISSN 0002-9939, FullText, MathReview (Jerzy Kaczorowski), ZentralBlatt
Referenced by:: (25.15.7), (25.15.8), §25.15(ii), §25.15.
Encodings:: BibTeX

…

10: 34.5 Basic Properties: $\mathit{6j}$ Symbol

… ►

34.5.11

{(2j_{1}+1)\left((J_{3}+J_{2}-J_{1})(L_{3}+L_{2}-J_{1})-2(J_{3}L_{3}+J_{2}L_{2% }-J_{1}L_{1})\right)\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}}\\ =j_{1}E(j_{1}+1)\begin{Bmatrix}j_{1}+1&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}+(j_{1}+1)E(j_{1})\begin{Bmatrix}j_{1}-1&j_{2}&j% _{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix},

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $l,l_{r}$ : non-negative integers or non-negative integers plus one half., $J_{r}$ , $L_{r}$ and $E(j)$
Permalink:: http://dlmf.nist.gov/34.5.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(iii), §34.5 and Ch.34

… ►

34.5.13

E(j)=\left((j^{2}-(j_{2}-j_{3})^{2})((j_{2}+j_{3}+1)^{2}-j^{2})(j^{2}-(l_{2}-l% _{3})^{2})((l_{2}+l_{3}+1)^{2}-j^{2})\right)^{\frac{1}{2}}.

ⓘ

Defines:: $E(j)$ (locally)
Symbols:: $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.5.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(iii), §34.5 and Ch.34

…