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1: Sidebar 5.SB1: Gamma & Digamma Phase Plots

… ►Phase changes around the zeros are of opposite sign to those around the poles. The fluid flow analogy in this case involves a line of vortices of alternating sign of circulation, resulting in a near cancellation of flow far from the real axis.

2: David M. Bressoud

… ► 227, in 1980, Factorization and Primality Testing, published by Springer-Verlag in 1989, Second Year Calculus from Celestial Mechanics to Special Relativity, published by Springer-Verlag in 1992, A Radical Approach to Real Analysis, published by the Mathematical Association of America in 1994, with a second edition in 2007, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, published by the Mathematical Association of America and Cambridge University Press in 1999, A Course in Computational Number Theory (with S. …

3: 14.27 Zeros

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

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4: 36.5 Stokes Sets

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36.5.7 $$X=\frac{9}{20}+20u^4-\frac{Y^2}{20u^2}+6u^2\mathrm{sign}\left(z\right),$$

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Symbols:

$\mathrm{sign}x$: sign of, $z$: real parameter, $X$: scaled coordinate and $Y$: scaled coordinate

Permalink:

http://dlmf.nist.gov/36.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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36.5.8 $$16u^5-\frac{Y^2}{10u}+4u^3\mathrm{sign}\left(z\right)-\frac{3}{10}|Y|\mathrm{sign}\left(z\right)+4t^5+2t^3\mathrm{sign}\left(z\right)+|Y|t^2=0,$$

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Symbols:

$\mathrm{sign}x$: sign of, $z$: real parameter, $t$: variable and $Y$: scaled coordinate

Permalink:

http://dlmf.nist.gov/36.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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36.5.9 $$t=-u+{\left(\frac{|Y|}{10u}-u^2-\frac{3}{10}\mathrm{sign}\left(z\right)\right)}^{1/2}.$$

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Symbols:

$\mathrm{sign}x$: sign of, $z$: real parameter, $t$: variable and $Y$: scaled coordinate

Referenced by:

§36.5(ii)

Permalink:

http://dlmf.nist.gov/36.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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5: 1.11 Zeros of Polynomials

… ►A monic polynomial of even degree with real coefficients has at least two zeros of opposite signs when the constant term is negative. ►

Descartes’ Rule of Signs

►The number of positive zeros of a polynomial with real coefficients cannot exceed the number of times the coefficients change sign, and the two numbers have same parity. A similar relation holds for the changes in sign of the coefficients of $f(-z)$, and hence for the number of negative zeros of $f(z)$. … ►Both polynomials have one change of sign; hence for each polynomial there is one positive zero, one negative zero, and six complex zeros. …

6: 4.16 Elementary Properties

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Table 4.16.1: Signs of the trigonometric functions in the four quadrants.

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Quadrant	$\sin \theta ,\csc \theta$	$\cos \theta ,\sec \theta$	$\tan \theta ,\cot \theta$
…

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Table 4.16.2: Trigonometric functions: quarter periods and change of sign.

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$x$	$-\theta$	$\frac{1}{2}\pi \pm \theta$	$\pi \pm \theta$	$\frac{3}{2}\pi \pm \theta$	$2\pi \pm \theta$
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7: 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: …

8: 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:

pMML, png, TeX

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:

pMML, png, TeX

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:

pMML, png, TeX

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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9: 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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10: 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. …