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11: 25.16 Mathematical Applications

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25.16.2 $$\psi \left(x\right)=x-\frac{{\zeta }^{\prime }\left(0\right)}{\zeta \left(0\right)}-\sum _{\rho }\frac{x^{\rho }}{\rho }+o\left(1\right),$$ $x\to \mathrm{\infty }$,

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Symbols:

$\psi \left(x\right)$: Chebyshev $\psi$-function, $\zeta \left(s\right)$: Riemann zeta function, $o\left(x\right)$: 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

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25.16.4 $$\psi \left(x\right)=x+O\left(x^{\frac{1}{2}+ϵ}\right),$$ $x\to \mathrm{\infty }$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\psi \left(x\right)$: Chebyshev $\psi$-function and $x$: real variable

Keywords:

asymptotic approximation

Source:

Apostol (2000, p. 9)

Permalink:

http://dlmf.nist.gov/25.16.E4

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 $s=1-2r$, $r=1,2,3,\mathrm{\dots }$. … ►

25.16.12 $$H(s,z)+H(z,s)=\zeta \left(s\right)\zeta \left(z\right)+\zeta \left(s+z\right),$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $H(s,z)$: generalized Euler sums, $s$: complex variable and $z$: complex variable

Keywords:

addition formula, connection formula

Source:

Apostol and Vu (1984, (11), p. 91)

Permalink:

http://dlmf.nist.gov/25.16.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §25.16(ii), §25.16 and Ch.25

…

12: 27.2 Functions

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

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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.

► ►►►

$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$
…
10	4	4	18	23	22	2	24	36	12	9	91	49	42	3	57
…

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13: 3.9 Acceleration of Convergence

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3.9.5 $$\ln 2=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\mathrm{\cdots }=\frac{1}{1\cdot 2^1}+\frac{1}{2\cdot 2^2}+\frac{1}{3\cdot 2^3}+\mathrm{\cdots },$$

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Symbols:

$\ln z$: principal branch of logarithm function

Permalink:

http://dlmf.nist.gov/3.9.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §3.9(ii), §3.9 and Ch.3

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3.9.6 $$\frac{\pi }{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\mathrm{\cdots }=\frac{1}{2}\left(1+\frac{1!}{1\cdot 3}+\frac{2!}{3\cdot 5}+\frac{3!}{3\cdot 5\cdot 7}+\mathrm{\cdots }\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/3.9.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §3.9(ii), §3.9 and Ch.3

… ►For further information on the epsilon algorithm see Brezinski and Redivo Zaglia (1991, pp. 78–95). … ►

Table 3.9.1: Shanks’ transformation for $s_n={\sum }_{j=1}^n{(-1)}^{j+1}{j}^{-2}$.

► ►►►►

$n$	$t_{n,2}$	$t_{n,4}$	$t_{n,6}$	$t_{n,8}$	$t_{n,10}$
…
8	0.82243 73137 33	0.82246 67719 32	0.82246 70301 49	0.82246 70333 73	0.82246 70334 23
9	0.82248 70624 89	0.82246 71865 91	0.82246 70351 34	0.82246 70334 48	0.82246 70334 24
…

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…

14: Bibliography P

… ►

$A=B$ . A K Peters Ltd., Wellesley, MA.

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Notes:

With a separately available computer disk

External Links:

ISBN 1-56881-063-6, MathReview (Peter Paule), ZentralBlatt

Referenced by:

§16.23(iv).

Encodings:

BibTeX

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Bounds for ratios of modified Bessel functions. Integral Transform. Spec. Funct. 9 (4), pp. 293–298.

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External Links:

ISSN 1065-2469, Document, MathReview, ZentralBlatt

Referenced by:

§10.37.

Encodings:

BibTeX

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On the maximum errors of polynomial approximations defined by interpolation and by least squares criteria. Comput. J. 9 (4), pp. 404–407.

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External Links:

Document, MathReview, ZentralBlatt

Referenced by:

§3.11(ii).

Encodings:

BibTeX

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Primzahlverteilung. Die Grundlehren der mathematischen Wissenschaften, Vol. 91, Springer-Verlag, Berlin-Göttingen-Heidelberg (German).

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Notes:

Reprinted in 1978.

External Links:

MathReview (A. E. Ingham), ZentralBlatt

Referenced by:

§27.11.

Encodings:

BibTeX

… ►

Elliptic integrals. Computers in Physics 4 (1), pp. 92–96.

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Referenced by:

2nd item.

Encodings:

BibTeX

…

15: Bibliography W

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Asymptotische Entwicklungen der Gaußschen hypergeometrischen Funktion für unbeschränkte Parameter. Z. Anal. Anwendungen 9 (4), pp. 351–360 (German).

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External Links:

ISSN 0232-2064, MathReview (T. Erber), ZentralBlatt

Referenced by:

§15.12(ii).

Encodings:

BibTeX

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Uniform asymptotic expansion of $J_{\nu }(\nu a)$ via a difference equation. Numer. Math. 91 (1), pp. 147–193.

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External Links:

ISSN 0029-599X, Document, MathReview (J. P. Singhal), ZentralBlatt

Referenced by:

§10.20(ii).

Encodings:

BibTeX

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Asymptotic approximations for certain $6$-$j$ and $9$-$j$ symbols. J. Phys. A 32 (39), pp. 6901–6902.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§34.8, §34.8.

Encodings:

BibTeX

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Surface Waves. In Handbuch der Physik, Vol. 9, Part 3, pp. 446–778.

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External Links:

Online edition, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§11.12.

Encodings:

BibTeX

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The Stokes set of the cusp diffraction catastrophe. J. Phys. A 13 (9), pp. 2913–2928.

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External Links:

ISSN 0305-4470, Document, MathReview (I. Stewart), ZentralBlatt

Referenced by:

§36.5(ii).

Encodings:

BibTeX

…

16: Bibliography V

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An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.

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External Links:

ISSN 0036-1410, Document, MathReview (William L. Perry), ZentralBlatt

Referenced by:

§31.10(i).

Encodings:

BibTeX

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Accurate calculation of prolate spheroidal radial functions of the first kind and their first derivatives. Quart. Appl. Math. 60 (3), pp. 589–599.

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Notes:

For software, see Van Buren (website)

External Links:

ISSN 0033-569X, MathReview (Alan M. Cohen), ZentralBlatt

Referenced by:

§30.16(iii).

Encodings:

BibTeX

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Integrating products of Bessel functions with an additional exponential or rational factor. Comput. Phys. Comm. 178 (8), pp. 578–590.

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Notes:

Associated computer program available online

External Links:

ISSN 0010-4655, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§10.74(vii), §10.74(vii).

Encodings:

BibTeX

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Dritter Beweis der die unvollständige Gammafunktion betreffenden Lochsschen Ungleichungen. Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 192 (1-3), pp. 83–91 (German).

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External Links:

ISSN 0723-9319, MathReview (Peter Schroth), ZentralBlatt

Referenced by:

§8.10.

Encodings:

BibTeX

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RFSFNS: A portable package for the numerical determination of the number and the calculation of roots of Bessel functions. Comput. Phys. Comm. 92 (2-3), pp. 252–266.

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Notes:

Double-precision Fortran, maximum accuracy 16S. For corrections see same journal 117 (1999), p. 290

External Links:

ISSN 0010-4655, Document, Code, ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

…

17: 23.10 Addition Theorems and Other Identities

… ►(23.10.8) continues to hold when $e_1$, $e_2$, $e_3$ are permuted cyclically. … ►For $n=2,3,\mathrm{\dots }$, …where … ►

23.10.17 $$\mathrm{\wp }\left(cz|c𝕃\right)=c^{-2}\mathrm{\wp }\left(z|𝕃\right),$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $𝕃$: lattice, $z$: complex and $c$: constant

A&S Ref:

18.2.2

Permalink:

http://dlmf.nist.gov/23.10.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §23.10(iv), §23.10 and Ch.23

… ►Also, when $𝕃$ is replaced by $c𝕃$ the lattice invariants $g_2$ and $g_3$ are divided by $c^4$ and $c^6$, respectively. …

18: 23.14 Integrals

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23.14.1 $$\int \mathrm{\wp }\left(z\right)dz=-\zeta \left(z\right),$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $dx$: differential of $x$ , $\int$: integral, $𝕃$: lattice and $z$: complex

A&S Ref:

18.6.1

Permalink:

http://dlmf.nist.gov/23.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §23.14 and Ch.23

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23.14.2 $$\int {\mathrm{\wp }}^2\left(z\right)dz=\frac{1}{6}{\mathrm{\wp }}^{\prime }\left(z\right)+\frac{1}{12}{g}_{2}z,$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $g_j$: Weierstrass lattice invariants $g_2$, $g_3$ , $dx$: differential of $x$ , $\int$: integral, $𝕃$: lattice and $z$: complex

A&S Ref:

18.7.1

Permalink:

http://dlmf.nist.gov/23.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.14 and Ch.23

►

23.14.3 $$\int {\mathrm{\wp }}^3\left(z\right)dz=\frac{1}{120}{\mathrm{\wp }}^{\prime \prime \prime }\left(z\right)-\frac{3}{20}{g}_2\zeta \left(z\right)+\frac{1}{10}{g}_{3}z.$$

ⓘ

Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $g_j$: Weierstrass lattice invariants $g_2$, $g_3$ , $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $dx$: differential of $x$ , $\int$: integral, $𝕃$: lattice and $z$: complex

A&S Ref:

18.7.2

Permalink:

http://dlmf.nist.gov/23.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.14 and Ch.23

…

19: 23.2 Definitions and Periodic Properties

… ►The generators of a given lattice $𝕃$ are not unique. …where $a,b,c,d$ are integers, then $2{\chi }_1$, $2{\chi }_3$ are generators of $𝕃$ iff … ►When $z\notin 𝕃$ the functions are related by … ►When it is important to display the lattice with the functions they are denoted by $\mathrm{\wp }\left(z|𝕃\right)$, $\zeta \left(z|𝕃\right)$, and $\sigma \left(z|𝕃\right)$, respectively. … ►If $2{\omega }_1$, $2{\omega }_3$ is any pair of generators of $𝕃$, and ${\omega }_2$ is defined by (23.2.1), then …

20: Bibliography F

… ►

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.

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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

… ►

Diffraction of radio waves around the earth’s surface. Acad. Sci. USSR. J. Phys. 9, pp. 255–266.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§9.1, Notations U, Notations V.

Encodings:

BibTeX

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Numerical evaluation of the elliptic integral of the third kind. Math. Comp. 19 (91), pp. 494–496.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§19.36(iv).

Encodings:

BibTeX

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Transverse cusp diffraction catastrophes produced by the reflection of ultrasonic tone bursts from a curved surface in water. J. Acoust. Soc. Amer. 92 (5), pp. 2869–2877.

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External Links:

Document

Referenced by:

§36.14(iv).

Encodings:

BibTeX

… ►

Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions. In Progress in Optics, E. Wolf (Ed.), Vol. 9, pp. 311–407.

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Referenced by:

§30.15(i), §30.15(ii), §30.15(iii), §30.15(iv), §30.15(v), §30.15(v).

Encodings:

BibTeX

…