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§5.4(iii) Extrema

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Table 5.4.1:

\Gamma'\left(x_{n}\right)=\psi\left(x_{n}\right)=0

.

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$n$	$x_{n}$	$\Gamma\left(x_{n}\right)$
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► …

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4.23.15

\operatorname{arccot}\left(-z\right)=-\operatorname{arccot}z,

z\neq\pm\mathrm{i}

.

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Symbols:: $\mathrm{i}$ : imaginary unit, $\operatorname{arccot}\NVar{z}$ : arccotangent function and $z$ : complex variable
A&S Ref:: 4.4.19 (Early printings had an error.)
Permalink:: http://dlmf.nist.gov/4.23.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §4.23(iii), §4.23 and Ch.4

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4.23.18

\operatorname{arccot}z=\pm\tfrac{1}{2}\pi-\operatorname{arctan}z,

\Re z\gtrless 0

.

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{arccot}\NVar{z}$ : arccotangent function, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\Re$ : real part and $z$ : complex variable
A&S Ref:: 4.4.5 (The first printing had an error.)
Referenced by:: §4.15(iii)
Permalink:: http://dlmf.nist.gov/4.23.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §4.23(iii), §4.23 and Ch.4

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Table 4.23.1: Inverse trigonometric functions: principal values at 0,

\pm 1

,

\pm\infty

.

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$x$	$\operatorname{arcsin}x$	$\operatorname{arccos}x$	$\operatorname{arctan}x$	$\operatorname{arccsc}x$	$\operatorname{arcsec}x$	$\operatorname{arccot}x$
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► ►For interrelations see Table 4.16.3. …

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4.40.14

\int\operatorname{arccsch}x\,\mathrm{d}x=x\operatorname{arccsch}x+% \operatorname{arcsinh}x,

0<x<\infty

,

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{arccsch}\NVar{z}$ : inverse hyperbolic cosecant function, $\operatorname{arcsinh}\NVar{z}$ : inverse hyperbolic sine function, $\int$ : integral and $x$ : real variable
A&S Ref:: 4.6.46 (modified. The first printing had an error.)
Permalink:: http://dlmf.nist.gov/4.40.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §4.40(iv), §4.40 and Ch.4

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4.40.15

\int\operatorname{arcsech}x\,\mathrm{d}x=x\operatorname{arcsech}x+% \operatorname{arcsin}x,

0<x<1

,

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{arcsech}\NVar{z}$ : inverse hyperbolic secant function, $\int$ : integral, $\operatorname{arcsin}\NVar{z}$ : arcsine function and $x$ : real variable
A&S Ref:: 4.6.47 (modified. The first printing had an error.)
Permalink:: http://dlmf.nist.gov/4.40.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §4.40(iv), §4.40 and Ch.4

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Equation (35.7.8)

Originally had the constraint $\Re\left(c\right),\Re\left(c-a-b\right)>\frac{1}{2}(m-1)$ . This constraint was replaced with $\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}$ ; ${\frac{1}{2}}(j+1)-a\in\mathbb{N}$ for some $j=1,\ldots,m$ ; ${\frac{1}{2}}(j+1)-c\notin\mathbb{N}$ and $c-a-b-{\frac{1}{2}}(m-j)\notin\mathbb{N}$ for all $j=1,\ldots,m$ .

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General

Several biographies had their publications updated.

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Graphics

A software bug that had corrupted some figures, such as those in About Color Map, has been corrected.

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Table 5.4.1

The table of extrema for the Euler gamma function $\Gamma$ had several entries in the $x_{n}$ column that were wrong in the last 2 or 3 digits. These have been corrected and 10 extra decimal places have been included.

$n$	$x_{n}$	$\Gamma\left(x_{n}\right)$
0	$1.46163\;21449\;68362\;34126$	$0.88560\;31944\;10888\;70028$
1	$-0.50408\;30082\;64455\;40926$	$-3.54464\;36111\;55005\;08912$
2	$-1.57349\;84731\;62390\;45878$	$2.30240\;72583\;39680\;13582$
3	$-2.61072\;08684\;44144\;65000$	$-0.88813\;63584\;01241\;92010$
4	$-3.63529\;33664\;36901\;09784$	$0.24512\;75398\;34366\;25044$
5	$-4.65323\;77617\;43142\;44171$	$-0.05277\;96395\;87319\;40076$
6	$-5.66716\;24415\;56885\;53585$	$0.00932\;45944\;82614\;85052$
7	$-6.67841\;82130\;73426\;74283$	$-0.00139\;73966\;08949\;76730$
8	$-7.68778\;83250\;31626\;03744$	$0.00018\;18784\;44909\;40419$
9	$-8.69576\;41638\;16401\;26649$	$-0.00002\;09252\;90446\;52667$
10	$-9.70267\;25400\;01863\;73608$	$0.00000\;21574\;16104\;52285$

Reported 2018-02-17 by David Smith.

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Paragraph Mellin–Barnes Integrals (in §8.6(ii))

The descriptions for the paths of integration of the Mellin-Barnes integrals (8.6.10)–(8.6.12) have been updated. The description for (8.6.11) now states that the path of integration is to the right of all poles. Previously it stated incorrectly that the path of integration had to separate the poles of the gamma function from the pole at $s=0$ . The paths of integration for (8.6.10) and (8.6.12) have been clarified. In the case of (8.6.10), it separates the poles of the gamma function from the pole at $s=a$ for $\gamma\left(a,z\right)$ . In the case of (8.6.12), it separates the poles of the gamma function from the poles at $s=0,1,2,\ldots$ .

Reported 2017-07-10 by Kurt Fischer.

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§33.24 Tables

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•

Abramowitz and Stegun (1964, Chapter 14) tabulates $F_{0}\left(\eta,\rho\right)$ , $G_{0}\left(\eta,\rho\right)$ , $F_{0}'\left(\eta,\rho\right)$ , and $G_{0}'\left(\eta,\rho\right)$ for $\eta=0.5(.5)20$ and $\rho=1(1)20$ , 5S; $C_{0}\left(\eta\right)$ for $\eta=0(.05)3$ , 6S.

… ►For earlier tables see Hull and Breit (1959) and Fletcher et al. (1962, §22.59).

§24.20 Tables

… ►Wagstaff (1978) gives complete prime factorizations of

N_{n}

and

E_{n}

for

n=20(2)60

and

n=8(2)42

, respectively. … ►For information on tables published before 1961 see Fletcher et al. (1962, v. 1, §4) and Lebedev and Fedorova (1960, Chapters 11 and 14).

§8.26 Tables

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§8.26(ii) Incomplete Gamma Functions

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§8.26(iii) Incomplete Beta Functions

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§8.26(iv) Generalized Exponential Integral

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•

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.

§6.19 Tables

… ►This section lists relevant tables that appeared later. … ►

•

Zhang and Jin (1996, pp. 652, 689) includes $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $x=0(.5)20(2)30$ , 8D; $\operatorname{Ei}\left(x\right)$ , $E_{1}\left(x\right)$ , $x=[0,100]$ , 8S.

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

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•

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.

List of Tables

§7.23 Tables

… ►Lebedev and Fedorova (1960) and Fletcher et al. (1962) give comprehensive indexes of mathematical tables. This section lists relevant tables that appeared later. … ►

•

Zhang and Jin (1996, pp. 637, 639) includes $(2/\sqrt{\pi})e^{-x^{2}}$ , $\operatorname{erf}x$ , $x=0(.02)1(.04)3$ , 8D; $C\left(x\right)$ , $S\left(x\right)$ , $x=0(.2)10(2)100(100)500$ , 8D.

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•

Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of $\operatorname{erf}z$ , $x\in[0,5]$ , $y=0.5(.5)3$ , 7D and 8D, respectively; the real and imaginary parts of $\int_{x}^{\infty}e^{\pm\mathrm{i}t^{2}}\,\mathrm{d}t$ , $(1/\sqrt{\pi})e^{\mp\mathrm{i}(x^{2}+(\pi/4))}\int_{x}^{\infty}e^{\pm\mathrm{i% }t^{2}}\,\mathrm{d}t$ , $x=0(.5)20(1)25$ , 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

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had%C3%A9%20table

11—20 of 237 matching pages

11: 5.4 Special Values and Extrema

§5.4(iii) Extrema

12: 4.23 Inverse Trigonometric Functions

13: 4.40 Integrals

14: Errata

15: 33.24 Tables

§33.24 Tables

16: 24.20 Tables

§24.20 Tables

17: 8.26 Tables