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11: 1.16 Distributions

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1.16.44 $$\mathrm{sign}\left(x\right)=2H\left(x\right)-1,$$ $x\ne 0$,

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Symbols:

$H\left(x\right)$: Heaviside function and $\mathrm{sign}x$: sign of

Permalink:

http://dlmf.nist.gov/1.16.E44

Encodings:

pMML, png, TeX

See also:

Annotations for §1.16(viii), §1.16 and Ch.1

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1.16.45 $${\mathrm{sign}}^{\prime }=2H^{\prime }=2\delta ,$$

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Symbols:

$H\left(x\right)$: Heaviside function, ${\delta }_x$: Dirac delta distribution and $\mathrm{sign}x$: sign of

Permalink:

http://dlmf.nist.gov/1.16.E45

Encodings:

pMML, png, TeX

See also:

Annotations for §1.16(viii), §1.16 and Ch.1

… ►and from (1.16.36) with $u=\mathrm{sign}$, $P(𝐃)=𝐷$, and $P_{-}(x)=-\mathrm{i}x$, we have also ►

1.16.47 $$ℱ\left({\mathrm{sign}}^{\prime }\right)=\frac{x}{\mathrm{i}}ℱ\left(\mathrm{sign}\right).$$

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Symbols:

$ℱ\left(u\right)$: Fourier transform of a tempered distribution, $\mathrm{i}$: imaginary unit and $\mathrm{sign}x$: sign of

Referenced by:

§1.16(viii)

Permalink:

http://dlmf.nist.gov/1.16.E47

Encodings:

pMML, png, TeX

See also:

Annotations for §1.16(viii), §1.16 and Ch.1

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1.16.48 $$ℱ\left(\mathrm{sign}\right)=\sqrt{\frac{2}{\pi }}\frac{\mathrm{i}}{x},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $ℱ\left(u\right)$: Fourier transform of a tempered distribution, $\mathrm{i}$: imaginary unit and $\mathrm{sign}x$: sign of

Referenced by:

§1.16(viii)

Permalink:

http://dlmf.nist.gov/1.16.E48

Encodings:

pMML, png, TeX

See also:

Annotations for §1.16(viii), §1.16 and Ch.1

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12: 15.13 Zeros

… ►where $S=\mathrm{sign}\left(\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-a\right)\mathrm{\Gamma }\left(c-b\right)\right)$. …

13: 26.15 Permutations: Matrix Notation

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26.15.1

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Permalink:

http://dlmf.nist.gov/26.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §26.15 and Ch.26

►The sign of the permutation $\sigma$ is the sign of the determinant of its matrix representation. …

14: 28.14 Fourier Series

… ►Ambiguities in sign are resolved by (28.14.9) when $q=0$, and by continuity for other values of $q$. … ►For changes of sign of $\nu$, $q$, and $m$, …

15: 28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

… ►The upper signs correspond to ${\mathrm{M}}_{\nu }^{(3)}(z,h)$ and the lower signs to ${\mathrm{M}}_{\nu }^{(4)}(z,h)$. …

16: 31.12 Confluent Forms of Heun’s Equation

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31.12.3 $$\frac{d^{2}w}{{dz}^2}-\left(\frac{\gamma }{z}+\delta +z\right)\frac{dw}{dz}+\frac{\alpha z-q}{z}w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $q$: real or complex parameter and $\alpha$: real or complex parameter

Referenced by:

Erratum (V1.0.7) for Equation (31.12.3)

Permalink:

http://dlmf.nist.gov/31.12.E3

Encodings:

pMML, png, TeX

Errata (effective with 1.0.7):

Originally the sign in front of the second term in this equation was $+$. The correct sign is $-$.

Reported 2013-10-31 by Henryk Witek

See also:

Annotations for §31.12, §31.12 and Ch.31

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17: Mathematical Introduction

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$(a,b]$ or $[a,b)$	half-closed intervals.
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$\mathrm{sign}x$	$-1$ if ; 0 if $x=0$; 1 if $x>0$.
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18: 3.1 Arithmetics and Error Measures

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3.1.1 $$x={(-1)}^s\cdot (b_0\mathrm{.}{b}_1{b}_2\mathrm{\dots }{b}_{p-1})\cdot 2^E,$$ $b_0=1$,

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Symbols:

$b_p$: bit, $s$: sign, $p$: number of significant bits and $E$: exponent

Permalink:

http://dlmf.nist.gov/3.1.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §3.1(i), §3.1 and Ch.3

►where $s$ is equal to $1$ or $0$, each $b_j$, $j\ge 1$, is either $0$ or $1$, $b_1$ is the most significant bit, $p$ ($\in \mathbb{N}$) is the number of significant bits $b_j$, $b_{p-1}$ is the least significant bit, $E$ is an integer called the exponent, $b_0.b_1{b}_2\mathrm{\dots }{b}_{p-1}$ is the significand, and $f=\mathrm{.}{b}_1{b}_2\mathrm{\dots }{b}_{p-1}$ is the fractional part. … ►

3.1.2 $${(-1)}^s{2}^E\sum _{j=0}^{p-1}{b}_j{2}^{-j},$$

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Symbols:

$b_p$: bit, $s$: sign, $p$: number of significant bits and $E$: exponent

Permalink:

http://dlmf.nist.gov/3.1.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §3.1(i), §3.1 and Ch.3

… ►For given values of $E_{\min }$, $E_{\max }$, and $p$, the format width in bits $N$ of a computer word is the total number of bits: the sign (one bit), the significant bits $b_1,b_2,\mathrm{\dots },b_{p-1}$ ($p-1$ bits), and the bits allocated to the exponent (the remaining $N-p$ bits). …

19: 4.23 Inverse Trigonometric Functions

… ►upper signs being taken on upper sides, and lower signs on lower sides. … ►the upper/lower signs corresponding to the upper/lower sides. … ►the upper/lower sign corresponding to the right/left side. … ►

4.23.34 $$\arcsin z=\arcsin \beta +\mathrm{i}\mathrm{sign}\left(y\right)\ln \left(\alpha +{({\alpha }^2-1)}^{1/2}\right),$$

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Symbols:

$[a,b]$: closed interval, $\in$: element of, $\mathrm{i}$: imaginary unit, $\arcsin z$: arcsine function, $\ln z$: principal branch of logarithm function, $\notin$: not an element of, $(a,b)$: open interval, $\mathrm{sign}x$: sign of, $x$: real variable, $y$: real variable, $z$: complex variable, $\alpha$ and $\beta$

A&S Ref:

4.4.37 (with the general value.)

Referenced by:

§4.23(vi), Erratum (V1.0.7) for Equations (4.23.34) and (4.23.35)

Permalink:

http://dlmf.nist.gov/4.23.E34

Encodings:

pMML, png, TeX

Errata (effective with 1.0.7):

Originally the factor $\mathrm{sign}\left(y\right)$ was missing from the second term on the right side of this equation. Also, the originally stated condition $x\in [-1,1]$ for this equation, stated on the line following (4.23.36), was replaced with the more general condition $\pm z\notin (1,\mathrm{\infty })$.

Reported 2013-07-01 by Volker Thürey

See also:

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

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4.23.35 $$\arccos z=\arccos \beta -\mathrm{i}\mathrm{sign}\left(y\right)\ln \left(\alpha +{({\alpha }^2-1)}^{1/2}\right),$$

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Symbols:

$[a,b]$: closed interval, $\in$: element of, $\mathrm{i}$: imaginary unit, $\arccos z$: arccosine function, $\ln z$: principal branch of logarithm function, $\notin$: not an element of, $(a,b)$: open interval, $\mathrm{sign}x$: sign of, $x$: real variable, $y$: real variable, $z$: complex variable, $\alpha$ and $\beta$

A&S Ref:

4.4.38 (with the general value.)

Referenced by:

§4.23(vi), Erratum (V1.0.7) for Equations (4.23.34) and (4.23.35)

Permalink:

http://dlmf.nist.gov/4.23.E35

Encodings:

pMML, png, TeX

Errata (effective with 1.0.7):

Originally the factor $\mathrm{sign}\left(y\right)$ was missing from the second term on the right side of this equation. Also, the originally stated condition $x\in [-1,1]$ for this equation, stated on the line following (4.23.36), was replaced with the more general condition $\pm z\notin (1,\mathrm{\infty })$.

Reported 2013-07-01 by Volker Thürey

See also:

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

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20: 19.14 Reduction of General Elliptic Integrals

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19.14.3 $${\int }_0^x\frac{dt}{\sqrt{1+t^4}}=\frac{\mathrm{sign}\left(x\right)}{2}F(\varphi ,k),$$ $\cos \varphi ={\displaystyle \frac{1-x^2}{1+x^2}}$, $k^2={\displaystyle \frac{1}{2}}$.

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Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\int$: integral, $\mathrm{sign}x$: sign of, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.14(i)

Permalink:

http://dlmf.nist.gov/19.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

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