sign function

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1: 32.11 Asymptotic Approximations for Real Variables

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32.11.9 $${\theta }_0=\frac{3}{2}{d}^2\ln 2+\mathrm{ph}\mathrm{\Gamma }\left(1-\frac{1}{2}\mathrm{i}{d}^2\right)+\frac{1}{4}\pi (1-2\mathrm{sign}\left(k\right)),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $\mathrm{sign}x$: sign of, $k$: real, $d$: constant and ${\theta }_0$: constant

Permalink:

http://dlmf.nist.gov/32.11.E9

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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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2: 1.16 Distributions

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1.16.39 $$ℱ\left(\delta \right)=\frac{1}{\sqrt{2\pi }},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${\delta }_x$: Dirac delta distribution and $ℱ\left(u\right)$: Fourier transform of a tempered distribution

Permalink:

http://dlmf.nist.gov/1.16.E39

Encodings:

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See also:

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

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

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1.16.46 $$ℱ\left({\mathrm{sign}}^{\prime }\right)=ℱ\left(2H^{\prime }\right)=2ℱ\left(\delta \right)=\sqrt{\frac{2}{\pi }},$$

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Symbols:

$H\left(x\right)$: Heaviside function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\delta }_x$: Dirac delta distribution, $ℱ\left(u\right)$: Fourier transform of a tempered distribution and $\mathrm{sign}x$: sign of

Referenced by:

§1.16(viii)

Permalink:

http://dlmf.nist.gov/1.16.E46

Encodings:

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

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1.16.49 $$⟨ℱ\left(\mathrm{sign}\right),\varphi ⟩=\mathrm{i}\sqrt{\frac{2}{\pi }}{⨍}_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\varphi (x)}{x}dx.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $ℱ\left(u\right)$: Fourier transform of a tempered distribution, $\mathrm{i}$: imaginary unit, ${⨍}_a^b$: Cauchy principal value, $\mathrm{sign}x$: sign of and $\varphi (x)$: test function

Permalink:

http://dlmf.nist.gov/1.16.E49

Encodings:

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See also:

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

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3: 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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4: 4.24 Inverse Trigonometric Functions: Further Properties

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4.24.4 $$\arctan z=\pm \frac{\pi }{2}-\frac{1}{z}+\frac{1}{3z^3}-\frac{1}{5z^5}+\mathrm{\cdots },$$ $\mathrm{\Re }z\gtrless 0$, $|z|\ge 1$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\arctan z$: arctangent function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.4.42 (has an error. $\frac{\pi }{2}$ should have a negative sign when .)

Referenced by:

§4.45(i)

Permalink:

http://dlmf.nist.gov/4.24.E4

Encodings:

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See also:

Annotations for §4.24(i), §4.24 and Ch.4

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5: 10.21 Zeros

… ► … ►For sign properties of the forward differences that are defined by … ► … ►The zeros of the functions … ►

§10.21(xiii) Rayleigh Function

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6: 10.44 Sums

… ►If $𝒵=I$ and the upper signs are taken, then the restriction on $\lambda$ is unnecessary. …

7: Bibliography S

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The sign of Lommel’s function. Trans. Amer. Math. Soc. 163, pp. 123–129.

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External Links:

ISSN 0002-9947, FullText, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§11.9(iv).

Encodings:

BibTeX

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8: 28.12 Definitions and Basic Properties

§28.12 Definitions and Basic Properties

… ►For change of signs of $\nu$ and $q$, … ►

§28.12(ii) Eigenfunctions ${\mathrm{me}}_{\nu }(z,q)$

… ►For changes of sign of $\nu$, $q$, and $z$, … ►For change of signs of $\nu$ and $z$, …

9: 4.37 Inverse Hyperbolic Functions

§4.37 Inverse Hyperbolic Functions

… ►the upper/lower signs corresponding to the right/left sides. … ►the upper/lower sign corresponding to the upper/lower side. … ►the upper/lower sign corresponding to the upper/lower side. … ►the upper/lower sign corresponding to the upper/lower sides. …

10: 33.2 Definitions and Basic Properties

… ►The function $F_{\mathrm{\ell }}(\eta ,\rho )$ is recessive (§2.7(iii)) at $\rho =0$, and is defined by …The choice of ambiguous signs in (33.2.3) and (33.2.4) is immaterial, provided that either all upper signs are taken, or all lower signs are taken. … ►The functions $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ are defined by … … ►

§33.2(iv) Wronskians and Cross-Product

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