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11: Errata

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The constraint which originally read “$\mathrm{\Re }s>0$, $\mathrm{\Re }a>0$, $z\in ℂ\setminus [1,\mathrm{\infty })$” has been extended to be “$\mathrm{\Re }s>1$, $\mathrm{\Re }a>0$ if $z=1$; $\mathrm{\Re }s>0$, $\mathrm{\Re }a>0$ if $z\in ℂ\setminus [1,\mathrm{\infty })$”.

Reported by Gergő Nemes on 2021-09-14

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In the paragraph immediately below (25.10.4), it was originally stated that “more than one-third of all zeros in the critical strip lie on the critical line.” which referred to Levinson (1974). This sentence has been updated with “one-third” being replaced with “41%” now referring to Bui et al. (2011) (suggested by Gergő Nemes on 2021-08-23).

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In the online version, the leading divided difference operators were previously omitted from these formulas, due to programming error.

Reported by Nico Temme on 2021-06-01

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Originally a minus sign was missing in the entries for $\mathrm{cd}u$ and $\mathrm{dc}u$ in the second column (headed $z+K+i{K}^{\prime }$). The correct entries are $-k^{-1}\mathrm{ns}z$ and $-k\mathrm{sn}z$. Note: These entries appear online but not in the published print edition. More specifically, Table 22.4.3 in the published print edition is restricted to the three Jacobian elliptic functions $\mathrm{sn},\mathrm{cn},\mathrm{dn}$, whereas Table 22.4.3 covers all 12 Jacobian elliptic functions.

	$u$
	$z+K$	$z+K+\mathrm{i}{K}^{\prime }$	$z+\mathrm{i}{K}^{\prime }$	$z+2K$	$z+2K+2\mathrm{i}{K}^{\prime }$	$z+2\mathrm{i}{K}^{\prime }$
$\mathrm{⋮}$
$\mathrm{cd}u$	$-\mathrm{sn}z$	$-k^{-1}\mathrm{ns}z$	$k^{-1}\mathrm{dc}z$	$-\mathrm{cd}z$	$-\mathrm{cd}z$	$\mathrm{cd}z$
$\mathrm{⋮}$
$\mathrm{dc}u$	$-\mathrm{ns}z$	$-k\mathrm{sn}z$	$k\mathrm{cd}z$	$-\mathrm{dc}z$	$-\mathrm{dc}z$	$\mathrm{dc}z$
$\mathrm{⋮}$

Reported 2014-02-28 by Svante Janson.

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Bibliographic citations were added in §§3.5(iv), 4.44, 8.22(ii), 22.4(i), and minor clarifications were made in §§19.12, 20.7(vii), 22.9(i). In addition, several minor improvements were made affecting only ancilliary documents and links in the online version.

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12: 30.6 Functions of Complex Argument

… ►of (30.2.1) with $\mu =m$ and $\lambda ={\lambda }_n^m\left({\gamma }^2\right)$ are real when $z\in (1,\mathrm{\infty })$, and their principal values (§4.2(i)) are obtained by analytic continuation to $ℂ\setminus (-\mathrm{\infty },1]$. …

13: 25.14 Lerch’s Transcendent

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25.14.5 $$\mathrm{\Phi }(z,s,a)=\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\frac{x^{s-1}{\mathrm{e}}^{-ax}}{1-z{\mathrm{e}}^{-x}}dx,$$ $\mathrm{\Re }s>1$, $\mathrm{\Re }a>0$ if $z=1$; $\mathrm{\Re }s>0$, $\mathrm{\Re }a>0$ if $z\in ℂ\setminus [1,\mathrm{\infty })$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{\Phi }(z,s,a)$: Lerch’s transcendent, $[a,b)$: half-closed interval, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part, $\setminus$: set subtraction, $x$: real variable, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Keywords:

improper integral, integral representation

Source:

Erdélyi et al. (1953a, (1.11.3), p. 27)

Referenced by:

Erratum (V1.1.4) for Equation (25.14.5)

Permalink:

http://dlmf.nist.gov/25.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §25.14(ii), §25.14 and Ch.25

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14: 25.12 Polylogarithms

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25.12.3 $${\mathrm{Li}}_2\left(z\right)+{\mathrm{Li}}_2\left(\frac{z}{z-1}\right)=-\frac{1}{2}{(\ln \left(1-z\right))}^2,$$ $z\in ℂ\setminus [1,\mathrm{\infty })$.

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Symbols:

$[a,b)$: half-closed interval, $ℂ$: complex plane, ${\mathrm{Li}}_2\left(z\right)$: dilogarithm, $\in$: element of, $\ln z$: principal branch of logarithm function, $\setminus$: set subtraction, $x$: real variable and $z$: complex variable

Keywords:

connection formula

Source:

Maximon (2003, (3.4), p. 2808)

A&S Ref:

27.7.5 (is a modified form with $z=1-x$)

Permalink:

http://dlmf.nist.gov/25.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(i), §25.12 and Ch.25

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25.12.4 $${\mathrm{Li}}_2\left(z\right)+{\mathrm{Li}}_2\left(\frac{1}{z}\right)=-\frac{1}{6}{\pi }^2-\frac{1}{2}{(\ln \left(-z\right))}^2,$$ $z\in ℂ\setminus [0,\mathrm{\infty })$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $[a,b)$: half-closed interval, $ℂ$: complex plane, ${\mathrm{Li}}_2\left(z\right)$: dilogarithm, $\in$: element of, $\ln z$: principal branch of logarithm function, $\setminus$: set subtraction and $z$: complex variable

Keywords:

connection formula

Source:

Maximon (2003, (3.2), p. 2808)

Referenced by:

§25.12(ii)

Permalink:

http://dlmf.nist.gov/25.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(i), §25.12 and Ch.25

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15: 19.2 Definitions

… ►Assume $1-{\sin }^2\varphi \in ℂ\setminus (-\mathrm{\infty },0]$ and $1-k^2{\sin }^2\varphi \in ℂ\setminus (-\mathrm{\infty },0]$, except that one of them may be 0, and $1-{\alpha }^2{\sin }^2\varphi \in ℂ\setminus \{0\}$. … ►If the line segment with endpoints $x$ and $y$ lies in $ℂ\setminus (-\mathrm{\infty },0]$, then …

16: 19.16 Definitions

… ►In (19.16.1)–(19.16.2_5), $x,y,z\in ℂ\setminus (-\mathrm{\infty },0]$ except that one or more of $x,y,z$ may be 0 when the corresponding integral converges. … ►

19.16.9 $$R_{-a}(𝐛;𝐳)=\frac{1}{\mathrm{B}(a,a^{\prime })}{\int }_0^{\mathrm{\infty }}{t}^{a^{\prime }-1}\prod _{j=1}^n{(t+z_j)}^{-b_j}dt=\frac{1}{\mathrm{B}(a,a^{\prime })}{\int }_0^{\mathrm{\infty }}{t}^{a-1}\prod _{j=1}^n{(1+t{z}_j)}^{-b_j}dt,$$ $b_1+\mathrm{\cdots }+b_n>a>0$, $b_j\in ℝ$, $z_j\in ℂ\setminus (-\mathrm{\infty },0]$,

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Defines:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function

Symbols:

$\mathrm{B}(a,b)$: beta function, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $(a,b]$: half-closed interval, $ℝ$: real line, $\setminus$: set subtraction and $n$: nonnegative integer

Referenced by:

§19.16(ii), §19.16(ii), §19.19, §19.20(iv), Erratum (V1.0.18) for Equation (19.16.9)

Permalink:

http://dlmf.nist.gov/19.16.E9

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.18):

The constraint $a,a^{\prime }>0$, was replaced with $b_1+\mathrm{\cdots }+b_n>a>0$, $b_j\in ℝ$. It therefore follows from (19.16.10) that $a^{\prime }>0$.

Suggested 2017-07-25 by Bastien Roucariès

See also:

Annotations for §19.16(ii), §19.16 and Ch.19

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17: 4.37 Inverse Hyperbolic Functions

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4.37.24 $$\mathrm{arctanh}z=\frac{1}{2}\ln \left(\frac{1+z}{1-z}\right),$$ $z\in ℂ\setminus (-\mathrm{\infty },-1]\cup [1,\mathrm{\infty })$;

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Symbols:

$[a,b)$: half-closed interval, $ℂ$: complex plane, $\in$: element of, $\mathrm{arctanh}z$: inverse hyperbolic tangent function, $\ln z$: principal branch of logarithm function, $(a,b]$: half-closed interval, $\setminus$: set subtraction, $\cup$: union and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

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18: 4.23 Inverse Trigonometric Functions

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4.23.26 $$\arctan z=\frac{\mathrm{i}}{2}\ln \left(\frac{\mathrm{i}+z}{\mathrm{i}-z}\right),$$ $z/\mathrm{i}\in ℂ\setminus (-\mathrm{\infty },-1]\cup [1,\mathrm{\infty })$;

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Symbols:

$[a,b)$: half-closed interval, $ℂ$: complex plane, $\in$: element of, $\mathrm{i}$: imaginary unit, $\arctan z$: arctangent function, $\ln z$: principal branch of logarithm function, $(a,b]$: half-closed interval, $\setminus$: set subtraction, $\cup$: union and $z$: complex variable

Referenced by:

§4.23(iv)

Permalink:

http://dlmf.nist.gov/4.23.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(iv), §4.23(iv), §4.23 and Ch.4

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19: 30.11 Radial Spheroidal Wave Functions

… ►When $z\in ℂ\setminus (-\mathrm{\infty },1]$ …

20: 19.25 Relations to Other Functions

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19.25.36 $$\mathrm{\wp }\left(z\right)-e_j\in ℂ\setminus (-\mathrm{\infty },0],$$ $j=1,2,3.$

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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, $e_j$: Weierstrass lattice roots, $ℂ$: complex plane, $\in$: element of, $(a,b]$: half-closed interval, $\setminus$: set subtraction, $j$: $(=1,2,3)$ and $z$: complex number

Referenced by:

(19.25.39)

Permalink:

http://dlmf.nist.gov/19.25.E36

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(vi), §19.25 and Ch.19

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