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21: 8.21 Generalized Sine and Cosine Integrals

… ►In these representations the integration paths do not cross the negative real axis, and in the case of (8.21.4) and (8.21.5) the paths also exclude the origin. …

22: 19.7 Connection Formulas

… ►If $k^2$ and ${\alpha }^2$ are real, then both integrals are circular cases or both are hyperbolic cases (see §19.2(ii)). …

23: 8.11 Asymptotic Approximations and Expansions

… ►in both cases uniformly with respect to bounded real values of $y$. …

24: 15.6 Integral Representations

… ►

15.6.1 $$𝐅(a,b;c;z)=\frac{1}{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-b\right)}{\int }_0^1\frac{t^{b-1}{(1-t)}^{c-b-1}}{{(1-zt)}^a}dt,$$ ; $\mathrm{\Re }c>\mathrm{\Re }b>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ 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 and $c$: real or complex parameter

Keywords:

Mellin transform

Referenced by:

§15.19(iii), (15.4.34), §15.6, §15.6, Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9), Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9)

Permalink:

http://dlmf.nist.gov/15.6.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.23):

The Olver hypergeometric function $𝐅(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.

See also:

Annotations for §15.6 and Ch.15

… ►In all cases the integrands are continuous functions of $t$ on the integration paths, except possibly at the endpoints. … ►In (15.6.3) the point $1/(z-1)$ lies outside the integration contour, the contour cuts the real axis between $t=-1$ and $0$, at which point $\mathrm{ph}t=\pi$ and $\mathrm{ph}\left(1+t\right)=0$. ►In (15.6.4) the point $1/z$ lies outside the integration contour, and at the point where the contour cuts the negative real axis $\mathrm{ph}t=\pi$ and $\mathrm{ph}\left(1-t\right)=0$. ►In (15.6.5) the integration contour starts and terminates at a point $A$ on the real axis between $0$ and $1$. …

25: 23.23 Tables

… ►2 in Abramowitz and Stegun (1964) gives values of $\mathrm{\wp }\left(z\right)$, ${\mathrm{\wp }}^{\prime }\left(z\right)$, and $\zeta \left(z\right)$ to 7 or 8D in the rectangular and rhombic cases, normalized so that ${\omega }_1=1$ and ${\omega }_3=\mathrm{i}a$ (rectangular case), or ${\omega }_1=1$ and ${\omega }_3=\frac{1}{2}+\mathrm{i}a$ (rhombic case), for $a$ = 1. …The values are tabulated on the real and imaginary $z$-axes, mostly ranging from 0 to 1 or $\mathrm{i}$ in steps of length 0. 05, and in the case of $\mathrm{\wp }\left(z\right)$ the user may deduce values for complex $z$ by application of the addition theorem (23.10.1). …

26: 25.2 Definition and Expansions

… ►

25.2.2 $$\zeta \left(s\right)=\frac{1}{1-2^{-s}}\sum _{n=0}^{\mathrm{\infty }}\frac{1}{{(2n+1)}^s},$$ $\mathrm{\Re }s>1$.

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part, $n$: nonnegative integer and $s$: complex variable

Keywords:

infinite series

Proof sketch:

Derivable from (25.2.1).

A&S Ref:

23.2.20 (is the special case with integer values of $s$)

Referenced by:

(25.11.11)

Permalink:

http://dlmf.nist.gov/25.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(ii), §25.2 and Ch.25

►

25.2.3 $$\zeta \left(s\right)=\frac{1}{1-2^{1-s}}\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^{n-1}}{n^s},$$ $\mathrm{\Re }s>0$.

ⓘ

Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part, $n$: nonnegative integer and $s$: complex variable

Keywords:

infinite series

Source:

Apostol (1976, (27), p. 292)

A&S Ref:

23.2.19 (is the special case with integer values of $s$)

Referenced by:

§25.11(x)

Permalink:

http://dlmf.nist.gov/25.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(ii), §25.2 and Ch.25

…

27: 6.7 Integral Representations

… ►The first integrals on the right-hand sides apply when ; the second ones when $\mathrm{\Re }z\ge 0$ and (in the case of (6.7.14)) $z\ne 0$. …

28: 10.21 Zeros

… ►

►►

Figure 10.21.1: Zeros $\bullet \bullet \bullet$ of $Y_n\left(nz\right)$ in $|\mathrm{ph}z|\le \pi$. Case $n=1$, $-1.6\le \mathrm{\Re }z\le 2.6$. Magnify

►

►►

Figure 10.21.2: Zeros $\bullet \bullet \bullet$ of $H_n^{(1)}\left(nz\right)$ in $|\mathrm{ph}z|\le \pi$. Case $n=1$, $-2.8\le \mathrm{\Re }z\le 1.4$. Magnify

►

►►

Figure 10.21.3: Zeros $\bullet \bullet \bullet$ of $Y_n\left(nz\right)$ in $|\mathrm{ph}z|\le \pi$. Case $n=5$, $-2.6\le \mathrm{\Re }z\le 1.6$. Magnify

►

►►

Figure 10.21.4: Zeros $\bullet \bullet \bullet$ of $H_n^{(1)}\left(nz\right)$ in $|\mathrm{ph}z|\le \pi$. Case $n=5$, $-2.6\le \mathrm{\Re }z\le 1.6$. Magnify

►

►►

Figure 10.21.5: Zeros $\bullet \bullet \bullet$ of $Y_n\left(nz\right)$ in $|\mathrm{ph}z|\le \pi$. Case $n=10$, $-2.3\le \mathrm{\Re }z\le 1.9$. Magnify

…

29: 4.13 Lambert $W$-Function

… ►In the case of $k=0$ and real $z$ the series converges for $z\ge \mathrm{e}$. …

30: Errata

… ►

The first paragraph has been rewritten to correct reported errors. The new version is reproduced here.

Let $p$ $(\ne 0)$ and $q$ be real constants and

4.43.1

$A={\left(-\frac{4}{3}p\right)}^{1/2},$

$B={\left(\frac{4}{3}p\right)}^{1/2}.$

The roots of

4.43.2 $$z^3+pz+q=0$$

are:

1. $A\sin a$, $A\sin \left(a+\frac{2}{3}\pi \right)$, and $A\sin \left(a+\frac{4}{3}\pi \right)$, with $\sin \left(3a\right)=4q/A^3$, when $4p^3+27q^2\le 0$.

2. $A\cosh a$, $A\cosh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $A\cosh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\cosh \left(3a\right)=-4q/A^3$, when , , and $4p^3+27q^2>0$.

3. $B\sinh a$, $B\sinh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $B\sinh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\sinh \left(3a\right)=-4q/B^3$, when $p>0$.

Note that in Case (a) all the roots are real, whereas in Cases (b) and (c) there is one real root and a conjugate pair of complex roots. See also §1.11(iii).

Reported 2014-10-31 by Masataka Urago.

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