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1: Bibliography I

… ►

Highly Oscillatory Quadrature: The Story So Far. In Numerical Mathematics and Advanced Applications, A. Bermudez de Castro and others (Eds.), pp. 97–118.

ⓘ

Notes:

Proceedings of ENuMath, Santiago de Compostela (2005)

External Links:

ISBN 3-540-34287-7, MathReview, ZentralBlatt

Referenced by:

§3.5(vii).

Encodings:

BibTeX

… ►

Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.

ⓘ

External Links:

ISBN 978-0-521-78201-2; 0-521-78201-5, MathReview Entry, ZentralBlatt

Referenced by:

§1.4(v), Profile Mourad E.H. Ismail, Erratum (V1.0.20) for Chapter 18 Orthogonal Polynomials.

Encodings:

BibTeX

►

Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.

ⓘ

Notes:

With two chapters by Walter Van Assche, With a foreword by Richard A. Askey, Corrected reprint of the 2005 original.

External Links:

ISBN 978-0-521-14347-9, MathReview Entry, ZentralBlatt

Referenced by:

§18.12, (18.15.4_5), (18.16.19), (18.16.20), (18.16.21), §18.17(ii), §18.18(vii), §18.19, (18.2.20), §18.2(vi), §18.2(xii), §18.2(xii), §18.2(x), §18.20(i), §18.20(ii), §18.21(ii), §18.22(i), §18.22(ii), §18.22(iii), §18.23, §18.25(i), §18.25(iii), §18.26(i), §18.26(iv), (18.27.4), §18.27(i), §18.27(ii), §18.27(iii), §18.27(v), §18.27(vi), §18.28(i), §18.28(ii), §18.28(iii), §18.28(v), §18.28(vi), §18.28(vii), §18.28(viii), §18.30(vi), §18.30(vii), §18.30(iii), §18.30(iv), §18.30, §18.30(vi), §18.34(i), §18.34(ii), §18.34(ii), §18.34(iii), (18.35.4), (18.35.5), (18.35.6), (18.35.6_2), (18.35.6_3), (18.35.6_4), (18.35.7), (18.35.8), §18.35(ii), §18.35(iii), §18.36(iii), §18.38(ii), §18.39(iv), §18.39(iv), §18.39(iv), (18.40.4), Profile Mourad E.H. Ismail, Erratum (V1.0.20) for Chapter 18 Orthogonal Polynomials.

Encodings:

BibTeX

…

2: 6.14 Integrals

… ►

6.14.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}{E}_1\left(t\right)dt=\frac{1}{a}\ln \left(1+a\right),$$ $\mathrm{\Re }a>-1$,

ⓘ

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\int$: integral, $\ln z$: principal branch of logarithm function and $\mathrm{\Re }$: real part

A&S Ref:

5.1.34 (special case of)

Permalink:

http://dlmf.nist.gov/6.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.14(i), §6.14 and Ch.6

… ►

6.14.4 $${\int }_0^{\mathrm{\infty }}{E}_1^2\left(t\right)dt=2\ln 2,$$

ⓘ

Symbols:

$dx$: differential of $x$ , $E_1\left(z\right)$: exponential integral, $\int$: integral and $\ln z$: principal branch of logarithm function

A&S Ref:

5.1.33

Permalink:

http://dlmf.nist.gov/6.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.14(ii), §6.14 and Ch.6

… ►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).

3: 15.7 Continued Fractions

… ►

15.7.1 $$\frac{𝐅(a,b;c;z)}{𝐅(a,b+1;c+1;z)}=t_0-\frac{u_{1}z}{t_1-}\frac{u_{2}z}{t_2-}\frac{u_{3}z}{t_3-}\mathrm{\cdots },$$

ⓘ

Symbols:

$𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ 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 $$\frac{𝐅(a,b;c;z)}{𝐅(a,b+1;c+1;z)}=v_0-\frac{w_1}{v_1-}\frac{w_2}{v_2-}\frac{w_3}{v_3-}\mathrm{\cdots },$$

ⓘ

Symbols:

$𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ 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 $$\frac{𝐅(a,b;c;z)}{𝐅(a+1,b+1;c+1;z)}=x_0+\frac{y_1}{x_1+}\frac{y_2}{x_2+}\frac{y_3}{x_3+}\mathrm{\cdots },$$

ⓘ

Symbols:

$𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ 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

…

4: 33.20 Expansions for Small $|ϵ|$

… ►where ►

33.20.4 $${𝖥}_k(\mathrm{\ell };r)=\sum _{p=2k}^{3k}{(2r)}^{(p+1)/2}{C}_{k,p}{J}_{2\mathrm{\ell }+1+p}\left(\sqrt{8r}\right),$$ $r>0$,

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, ${𝖥}_k(\mathrm{\ell };r)$: coefficient and $C_{k,p}$: coefficients

Permalink:

http://dlmf.nist.gov/33.20.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §33.20(ii), §33.20 and Ch.33

… ►The functions $J$ and $I$ are as in §§10.2(ii), 10.25(ii), and the coefficients $C_{k,p}$ are given by $C_{0,0}=1$, $C_{1,0}=0$, and … ►where $A(ϵ,\mathrm{\ell })$ is given by (33.14.11), (33.14.12), and …The functions $Y$ and $K$ are as in §§10.2(ii), 10.25(ii), and the coefficients $C_{k,p}$ are given by (33.20.6). …

5: 34.5 Basic Properties: $\mathit{6}j$ Symbol

… ►

34.5.1

ⓘ

Symbols:

: $\mathit{6}j$ symbol, $j,j_r$: non-negative integers or non-negative integers plus one half. and $J$: sum

Permalink:

http://dlmf.nist.gov/34.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §34.5(i), §34.5 and Ch.34

►

34.5.2

ⓘ

Symbols:

: $\mathit{6}j$ symbol, $j,j_r$: non-negative integers or non-negative integers plus one half. and $J$: sum

Permalink:

http://dlmf.nist.gov/34.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §34.5(i), §34.5 and Ch.34

►

34.5.3

ⓘ

Symbols:

: $\mathit{6}j$ symbol, $j,j_r$: non-negative integers or non-negative integers plus one half. and $J$: sum

Permalink:

http://dlmf.nist.gov/34.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §34.5(i), §34.5 and Ch.34

►

34.5.4

ⓘ

Symbols:

: $\mathit{6}j$ symbol, $j,j_r$: non-negative integers or non-negative integers plus one half. and $J$: sum

Permalink:

http://dlmf.nist.gov/34.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §34.5(i), §34.5 and Ch.34

… ►For further recursion relations see Varshalovich et al. (1988, §9.6) and Edmonds (1974, pp. 98–99). …

6: Bibliography O

… ►

Summing one- and two-dimensional series related to the Euler series. J. Comput. Appl. Math. 98 (2), pp. 245–271.

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview (Boris Petrovich Osilenker), ZentralBlatt

Referenced by:

§25.8.

Encodings:

BibTeX

… ►

Hyperterminants. II. J. Comput. Appl. Math. 89 (1), pp. 87–95.

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§2.11(v).

Encodings:

BibTeX

… ►

Uniform asymptotic expansions for hypergeometric functions with large parameters. III. Analysis and Applications (Singapore) 8 (2), pp. 199–210.

ⓘ

External Links:

ISSN 0219-5305, Document, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

BibTeX

… ►

On an asymptotic expansion of a ratio of gamma functions. Proc. Roy. Irish Acad. Sect. A 95 (1), pp. 5–9.

ⓘ

External Links:

ISSN 0035-8975, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§5.11(iii).

Encodings:

BibTeX

… ►

Numerical evaluation of the dilogarithm of complex argument. Celestial Mech. Dynam. Astronom. 62 (1), pp. 93–98.

ⓘ

External Links:

ISSN 0923-2958, Document, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

…

7: 25.5 Integral Representations

… ►

25.5.7 $$\zeta \left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\sum _{m=1}^n\frac{B_{2m}}{(2m)!}{\left(s\right)}_{2m-1}+\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\left(\frac{1}{{\mathrm{e}}^x-1}-\frac{1}{x}+\frac{1}{2}-\sum _{m=1}^n\frac{B_{2m}}{(2m)!}{x}^{2m-1}\right)\frac{x^{s-1}}{{\mathrm{e}}^x}dx,$$ $\mathrm{\Re }s>-(2n+1)$, $n=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\mathrm{\Re }$: real part, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

improper integral, integral representation

Proof sketch:

Derivable from (25.5.6) by adding and subtracting ${\sum }_{m=1}^n{B}_{2m}{x}^{2m-1}/(2m)!$ in the integrand, using (5.2.1), (5.2.5), and recognizing that ${({\mathrm{e}}^x-1)}^{-1}-x^{-1}+1/2-{\sum }_{m=1}^n{B}_{2m}{x}^{2m-1}/(2m)!=O\left(x^{2n+1}\right)$ as $x\to 0$, demonstrating the region of convergence.

Referenced by:

Erratum (V1.0.9) for Chapters 7, 25

Permalink:

http://dlmf.nist.gov/25.5.E7

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.9):

We have rewritten the original summation ${\sum }_{m=1}^n\frac{B_{2m}}{(2m)!}\frac{\mathrm{\Gamma }\left(s+2m-1\right)}{\mathrm{\Gamma }\left(s\right)}$ more concisely as ${\sum }_{m=1}^n\frac{B_{2m}}{(2m)!}{\left(s\right)}_{2m-1}$ using the Pochhammer symbol.

See also:

Annotations for §25.5(i), §25.5 and Ch.25

… ►

25.5.10 $$\zeta \left(s\right)=\frac{2^{s-1}}{1-2^{1-s}}{\int }_0^{\mathrm{\infty }}\frac{\cos \left(s\arctan x\right)}{{(1+x^2)}^{s/2}\cosh \left(\frac{1}{2}\pi x\right)}dx.$$

ⓘ

Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, $\arctan z$: arctangent function, $x$: real variable and $s$: complex variable

Keywords:

improper integral, integral representation

Sources:

Lindelöf (1905, (6), p. 98); with $x=2t$; Erdélyi et al. (1953a, (1.12.15), p. 33)

Permalink:

http://dlmf.nist.gov/25.5.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §25.5(i), §25.5 and Ch.25

… ►

25.5.12 $$\zeta \left(s\right)=\frac{2^{s-1}}{s-1}-2^s{\int }_0^{\mathrm{\infty }}\frac{\sin \left(s\arctan x\right)}{{(1+x^2)}^{s/2}({\mathrm{e}}^{\pi x}+1)}dx.$$

ⓘ

Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\arctan z$: arctangent function, $\sin z$: sine function, $x$: real variable and $s$: complex variable

Keywords:

improper integral, integral representation

Source:

Lindelöf (1905, (4), p. 98); with $x=2t$

Permalink:

http://dlmf.nist.gov/25.5.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §25.5(i), §25.5 and Ch.25

… ►For ${\theta }_3$ see §20.2(i). … ►The contour here is any loop that encircles the origin in the positive direction not enclosing any of the points $\pm \pi \mathrm{i}$, $\pm 3\pi \mathrm{i}$, ….

8: 27.2 Functions

… ►

27.2.9 $$d\left(n\right)=\sum _{d|n}1$$

ⓘ

Symbols:

$d_k\left(n\right)$: divisor function, $d$: positive integer and $n$: positive integer

A&S Ref:

24.3.3 I.A

Permalink:

http://dlmf.nist.gov/27.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►It is the special case $k=2$ of the function $d_k\left(n\right)$ that counts the number of ways of expressing $n$ as the product of $k$ factors, with the order of factors taken into account. …Note that ${\sigma }_0\left(n\right)=d\left(n\right)$. … ►Table 27.2.2 tabulates the Euler totient function $\varphi \left(n\right)$, the divisor function $d\left(n\right)$ ($={\sigma }_0(n)$), and the sum of the divisors $\sigma (n)$ ($={\sigma }_1(n)$), for $n=1(1)52$. … ►

Table 27.2.2: Functions related to division.

► ►►►

$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$
…
13	12	2	14	26	12	4	42	39	24	4	56	52	24	6	98

►

9: Bibliography N

… ►

Algorithm 707: CONHYP: A numerical evaluator of the confluent hypergeometric function for complex arguments of large magnitudes. ACM Trans. Math. Software 18 (3), pp. 345–349.

ⓘ

Notes:

Double-precision Fortran, minimum accuracy 9S, maximum accuracy 13S.

External Links:

ISSN 0098-3500, Document, GAMS (module 707; package TOMS), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

… ►

On an asymptotic expansion of the Kontorovich-Lebedev transform. Methods Appl. Anal. 3 (1), pp. 98–108.

ⓘ

External Links:

ISSN 1073-2772, FullText, MathReview (C. M. Joshi), ZentralBlatt

Referenced by:

§10.43(v).

Encodings:

BibTeX

… ►

Generalization of Binet’s Gamma function formulas. Integral Transforms Spec. Funct. 24 (8), pp. 597–606.

ⓘ

External Links:

ISSN 1065-2469, Document, FullText, MathReview (Daniele Ritelli), ZentralBlatt

Referenced by:

§5.11(i), §5.11(i).

Encodings:

BibTeX

… ►

The asymptotic behavior of the general real solution of the third Painlevé equation. Dokl. Akad. Nauk SSSR 283 (5), pp. 1161–1165 (Russian).

ⓘ

Notes:

In Russian. English translation: Sov. Math., Dokl. 30(1988), no. 8, pp. 666–668.

External Links:

ISSN 0002-3264, MathReview (Vojislav Marić), ZentralBlatt

Referenced by:

§32.11(iv).

Encodings:

BibTeX

… ►

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

…

10: 10.60 Sums

… ►

10.60.14 $$\sum _{n=0}^{\mathrm{\infty }}(2n+1){({𝗃}_n^{\prime }\left(z\right))}^2=\frac{1}{3}.$$

ⓘ

Symbols:

${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

Referenced by:

§10.60(iii)

Permalink:

http://dlmf.nist.gov/10.60.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §10.60(iii), §10.60 and Ch.10

… ►For collections of sums of series relevant to spherical Bessel functions or Bessel functions of half odd integer order see Erdélyi et al. (1953b, pp. 43–45 and 98–105), Gradshteyn and Ryzhik (2000, §§8.51, 8.53), Hansen (1975), Magnus et al. (1966, pp. 106–108 and 123–138), and Prudnikov et al. (1986b, pp. 635–637 and 651–700). …