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21: 1.4 Calculus of One Variable

… ►where the sum is over all nonnegative integers $m_1,m_2,\mathrm{\dots },m_n$ that satisfy $m_1+2m_2+\mathrm{\cdots }+n{m}_n=n$, and $k=m_1+m_2+\mathrm{\cdots }+m_n$. … ►

Infinite Integrals

… ►Definite integrals over the Stieltjes measure $d\alpha (x)$ could represent a sum, an integral, or a combination of the two. … ►With , the total variation of $f(x)$ on a finite or infinite interval $(a,b)$ is …where the supremum is over all sets of points in the closure of $(a,b)$, that is, $(a,b)$ with $a,b$ added when they are finite. …

22: 2.11 Remainder Terms; Stokes Phenomenon

… ►Secondly, the asymptotic series represents an infinite class of functions, and the remainder depends on which member we have in mind. … ►uniformly when $\theta \in [-\pi +\delta ,\pi -\delta ]$ ($\delta >0$) and $|\alpha |$ is bounded. … ►Where should the change-over take place? Can it be accomplished smoothly? … ►Rays (or curves) on which one contribution in a compound asymptotic expansion achieves maximum dominance over another are called Stokes lines ($\theta =\pi$ in the present example). … ►For example, using double precision $d_{20}$ is found to agree with (2.11.31) to 13D. …

23: 6.19 Tables

… ►

Zhang and Jin (1996, pp. 652, 689) includes $\mathrm{Si}\left(x\right)$, $\mathrm{Ci}\left(x\right)$, $x=0(.5)20(2)30$, 8D; $\mathrm{Ei}\left(x\right)$, $E_1\left(x\right)$, $x=[0,100]$, 8S.

… ►

Abramowitz and Stegun (1964, Chapter 5) includes the real and imaginary parts of $z{\mathrm{e}}^z{E}_1\left(z\right)$, $x=-19(1)20$, $y=0(1)20$, 6D; ${\mathrm{e}}^z{E}_1\left(z\right)$, $x=-4(.5)-2$, $y=0(.2)1$, 6D; $E_1\left(z\right)+\ln z$, $x=-2(.5)2.5$, $y=0(.2)1$, 6D.

►

Zhang and Jin (1996, pp. 690–692) includes the real and imaginary parts of $E_1\left(z\right)$, $\pm x=0.5,1,3,5,10,15,20,50,100$, $y=0(.5)1(1)5(5)30,50,100$, 8S.

24: 10.43 Integrals

… ►

§10.43(ii) Integrals over the Intervals $(0,x)$ and $(x,\mathrm{\infty })$

… ►

§10.43(iv) Integrals over the Interval ($0,\mathrm{\infty }$)

… ►For the second equation there is a cut in the $a$-plane along the interval $[0,1]$, and all quantities assume their principal values (§4.2(i)). … ►For infinite integrals of triple products of modified and unmodified Bessel functions, see Gervois and Navelet (1984, 1985a, 1985b, 1986a, 1986b). …

25: 8.26 Tables

… ►

Khamis (1965) tabulates $P(a,x)$ for $a=0.05(.05)10(.1)20(.25)70$, $0.0001\le x\le 250$ to 10D.

… ►

Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_n\left(x\right)$ for $n=2,3,4,10,20$, $x=0(.01)2$ to 7D; also $(x+n){\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2,3,4,10,20$, $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

… ►

Pagurova (1961) tabulates $E_n\left(x\right)$ for $n=0(1)20$, $x=0(.01)2(.1)10$ to 4-9S; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ for $p=0(.1)1$, $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

… ►

Zhang and Jin (1996, Table 19.1) tabulates $E_n\left(x\right)$ for $n=1,2,3,5,10,15,20$, $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

26: 23 Weierstrass Elliptic and Modular
Functions

…

27: 25.11 Hurwitz Zeta Function

… ►

25.11.3 $$\zeta (s,a)=\zeta (s,a+1)+a^{-s},$$

ⓘ

Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $n$: nonnegative integer, $a$: real or complex parameter and $s$: complex variable

Keywords:

recurrence

Proof sketch:

Derivable from (25.11.1) by comparing its $n=0$ term with its sum over $n\ge 1$.

Permalink:

http://dlmf.nist.gov/25.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(i), §25.11 and Ch.25

►

25.11.4 $$\zeta (s,a)=\zeta (s,a+m)+\sum _{n=0}^{m-1}\frac{1}{{(n+a)}^s},$$ $m=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $m$: nonnegative integer, $n$: nonnegative integer, $a$: real or complex parameter and $s$: complex variable

Keywords:

recurrence

Proof sketch:

Derivable from (25.11.1) by comparing its sum over $n=0,\mathrm{\dots },m-1$, $m\ge 1$, with its sum over $n\ge m$.

Permalink:

http://dlmf.nist.gov/25.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(i), §25.11 and Ch.25

… ►

►►

Figure 25.11.1: Hurwitz zeta function $\zeta (x,a)$, $a$ = 0. …8, 1, $-20\le x\le 10$. … Magnify

… ►

25.11.35 $$\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n}{{(n+a)}^s}=\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\frac{x^{s-1}{\mathrm{e}}^{-ax}}{1+{\mathrm{e}}^{-x}}dx=2^{-s}\left(\zeta (s,\frac{1}{2}a)-\zeta (s,\frac{1}{2}(1+a))\right),$$ $\mathrm{\Re }a>0$, $\mathrm{\Re }s>0$; or $\mathrm{\Re }a=0$, $\mathrm{\Im }a\ne 0$, .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $\int$: integral, $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

improper integral, infinite series, integral representation, series representation

Proof sketch:

The infinite series connection to the difference of Hurwitz zeta functions is a restatement of (25.11.8). The integral representation is derivable from the difference of Hurwitz zeta functions by substituting (25.11.25) in each, making a change of variables $x=2t$, factoring, and expressing $1-{\mathrm{e}}^{-2t}=(1+{\mathrm{e}}^{-t})(1-{\mathrm{e}}^{-t})$.

Referenced by:

(25.11.31), §25.11(x), §25.11(x)

Permalink:

http://dlmf.nist.gov/25.11.E35

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(x), §25.11 and Ch.25

… ►

25.11.39 $$\sum _{k=2}^{\mathrm{\infty }}\frac{k}{2^k}\zeta (k+1,\frac{3}{4})=8G,$$

ⓘ

Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $k$: nonnegative integer and $G$: Catalan’s constant

Keywords:

Catalan’s constant, infinite series

Source:

Adamchik and Srivastava (1998, (2.30), p. 138)

Permalink:

http://dlmf.nist.gov/25.11.E39

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(xi), §25.11 and Ch.25

…

28: 26.12 Plane Partitions

… ►

Table 26.12.1: Plane partitions.

► ►►►

$n$	$\mathrm{pp}\left(n\right)$	$n$	$\mathrm{pp}\left(n\right)$	$n$	$\mathrm{pp}\left(n\right)$
…
3	6	20	75278	37	903 79784
…

►

… ►A plane partition is transpose complement if it is equal to the reflection through the $(x,y)$-plane of its complement. … ►The notation ${\sum }_{\pi \subseteq B(r,s,t)}$ denotes the sum over all plane partitions contained in $B(r,s,t)$, and $|\pi |$ denotes the number of elements in $\pi$. …

29: Bibliography B

… ►

Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

ⓘ

External Links:

Document

Referenced by:

§33.22(iv).

Encodings:

BibTeX

… ►

A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.

ⓘ

Notes:

Double-precision Fortran, maximum accuracy 10S.

External Links:

Document, Code, ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

… ►

Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

ⓘ

Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

1st item, 4th item.

Encodings:

BibTeX

… ►

Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.

ⓘ

External Links:

Document, ZentralBlatt

Referenced by:

§10.73(iv).

Encodings:

BibTeX

… ►

Rainbow over Woolsthorpe Manor. Notes and Records Roy. Soc. London 36 (1), pp. 3–11 (1 plate).

ⓘ

External Links:

ISSN 0035-9149, Document, MathReview

Referenced by:

Sidebar 9.SB1.

Encodings:

BibTeX

…

30: 28.30 Expansions in Series of Eigenfunctions

… ►Then every continuous $2\pi$-periodic function $f(x)$ whose second derivative is square-integrable over the interval $[0,2\pi ]$ can be expanded in a uniformly and absolutely convergent series …