Descartes’ rule of signs

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21: 25.10 Zeros

… ►Because $|Z(t)|=|\zeta \left(\frac{1}{2}+\mathrm{i}t\right)|$, $Z(t)$ vanishes at the zeros of $\zeta \left(\frac{1}{2}+\mathrm{i}t\right)$, which can be separated by observing sign changes of $Z(t)$. Because $Z(t)$ changes sign infinitely often, $\zeta \left(\frac{1}{2}+\mathrm{i}t\right)$ has infinitely many zeros with $t$ real. … ►By comparing $N(T)$ with the number of sign changes of $Z(t)$ we can decide whether $\zeta \left(s\right)$ has any zeros off the line in this region. Sign changes of $Z(t)$ are determined by multiplying (25.9.3) by $\exp \left(\mathrm{i}\vartheta (t)\right)$ to obtain the Riemann–Siegel formula: …

22: 32.11 Asymptotic Approximations for Real Variables

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If , then $w(x)$ changes sign once, from positive to negative, as $x$ passes from $x_0$ to $0$.

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32.11.10 $$w_k(x)\sim \mathrm{sign}\left(k\right)\sqrt{\frac{1}{2}|x|},$$ $x\to -\mathrm{\infty }$.

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Symbols:

$\sim$: asymptotic equality, $\mathrm{sign}x$: sign of, $x$: real and $k$: real

Permalink:

http://dlmf.nist.gov/32.11.E10

Encodings:

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See also:

Annotations for §32.11(ii), §32.11 and Ch.32

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32.11.11 $$w_k(x)\sim \mathrm{sign}\left(k\right){(x-c_0)}^{-1},$$ $x\to c_0+$.

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Symbols:

$\sim$: asymptotic equality, $\mathrm{sign}x$: sign of, $x$: real, $k$: real and $c_0$: pole

Permalink:

http://dlmf.nist.gov/32.11.E11

Encodings:

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See also:

Annotations for §32.11(ii), §32.11 and Ch.32

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32.11.17 $$d^2={\pi }^{-1}\ln \left(1+k^2\right),$$ $\mathrm{sign}\left(k\right)={(-1)}^n$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{sign}x$: sign of, $n$: integer, $k$: real and $d$: constant

Permalink:

http://dlmf.nist.gov/32.11.E17

Encodings:

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See also:

Annotations for §32.11(iii), §32.11 and Ch.32

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32.11.21 $$\sigma =-\mathrm{sign}\left(\mathrm{\Im }s\right),$$

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Symbols:

$\mathrm{\Im }$: imaginary part, $\mathrm{sign}x$: sign of, $\sigma$: real constant and $s$

Permalink:

http://dlmf.nist.gov/32.11.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §32.11(iii), §32.11 and Ch.32

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23: 34.7 Basic Properties: $\mathit{9}j$ Symbol

… ►This equation is the sum rule. It constitutes an addition theorem for the $\mathit{9}j$ symbol. …

24: 14.16 Zeros

… ► ${𝖰}_{\nu }^{\mu }\left(x\right)$ has $\max (\lceil \nu -|\mu |\rceil ,0)+k$ zeros in the interval $(-1,1)$, where $k$ can take one of the values $-1$, $0$, $1$, $2$, subject to $\max (\lceil \nu -|\mu |\rceil ,0)+k$ being even or odd according as $\cos \left(\nu \pi \right)$ and $\cos \left(\mu \pi \right)$ have opposite signs or the same sign. … ►

$\mu >0$, $\mu >\nu$, $\mu \notin \mathbb{Z}$, and $\sin \left((\mu -\nu )\pi \right)$ and $\sin \left(\mu \pi \right)$ have opposite signs.

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25: 26.13 Permutations: Cycle Notation

… ►For the example (26.13.2), this decomposition is given by $\left(1,3,2,5,7\right)\left(6,8\right)=\left(1,3\right)\left(2,3\right)\left(2,5\right)\left(5,7\right)\left(6,8\right).$ … ►The sign of a permutation is $+$ if the permutation is even, $-$ if it is odd. …

26: 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:

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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:

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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:

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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:

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See also:

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

…

27: 15.13 Zeros

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

28: 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:

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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. …

29: 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 $m$, …

30: 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 ${\mathrm{M}}_{\nu }^{(4)}(z,h)$. …