sign function

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21: 8.6 Integral Representations

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§8.6(i) Integrals Along the Real Line

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§8.6(ii) Contour Integrals

… ► …where the integration path passes above or below the pole at $t=1$, according as upper or lower signs are taken. … ►

§8.6(iii) Compendia

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22: 4.2 Definitions

… ► … ►with either upper signs or lower signs taken throughout. … ►

§4.2(iii) The Exponential Function

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§4.2(iv) Powers

… ►Again, without the closed definition the $\ge$ and $\le$ signs would have to be replaced by $>$ and , respectively.

23: 18.39 Applications in the Physical Sciences

… ►(where the minus sign is often omitted, as it arises as an arbitrary phase when taking the square root of the real, positive, norm of the wave function), allowing equation (18.39.37) to be rewritten in terms of the associated Coulomb–Laguerre polynomials ${𝐋}_{n+l}^{2l+1}({\rho }_n)$. … ►For either sign of $Z$, and $s$ chosen such that $n+l+1+(2Z/s)>0$, $n=0,1,2,\mathrm{\dots }$, truncation of the basis to $N$ terms, with $x_i^N\in [-1,1]$, the discrete eigenvectors are the orthonormal $L^2$ functions …

24: 4.23 Inverse Trigonometric Functions

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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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25: 5.11 Asymptotic Expansions

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5.11.3 $$\mathrm{\Gamma }\left(z\right)={\mathrm{e}}^{-z}{z}^z{\left(\frac{2\pi }{z}\right)}^{1/2}{\mathrm{\Gamma }}^{\ast }\left(z\right)\sim {\mathrm{e}}^{-z}{z}^z{\left(\frac{2\pi }{z}\right)}^{1/2}\sum _{k=0}^{\mathrm{\infty }}\frac{g_k}{z^k},$$

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Defines:

${\mathrm{\Gamma }}^{\ast }\left(z\right)$: scaled gamma function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer, $z$: complex variable and $g_k$: coefficients

A&S Ref:

6.1.37

Referenced by:

§10.19(i), §13.8(iv), §5.11(i), §5.11(i), §5.11(i), §5.11(ii), §5.11(ii), §5.21, §5.9(i), §8.11(ii), §8.12, (8.18.13), (8.18.13), §8.18(ii), §8.18(ii), Erratum (V1.1.7) for Additions, Erratum (V1.1.8) for Rearrangement

Permalink:

http://dlmf.nist.gov/5.11.E3

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

The definition of the scaled gamma function ${\mathrm{\Gamma }}^{\ast }\left(z\right)$ was inserted after the first equals sign.

Changes (effective with 1.0.11):

An unnecessary bracket around the sum was removed.

See also:

Annotations for §5.11(i), §5.11 and Ch.5

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26: 10.61 Definitions and Basic Properties

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§10.61(i) Definitions

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§10.61(ii) Differential Equations

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§10.61(iii) Reflection Formulas for Arguments

►In general, Kelvin functions have a branch point at $x=0$ and functions with arguments $x{\mathrm{e}}^{\pm \pi \mathrm{i}}$ are complex. … ►

§10.61(iv) Reflection Formulas for Orders

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27: 9.6 Relations to Other Functions

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9.6.3 $${\mathrm{Ai}}^{\prime }\left(z\right)=-{\pi }^{-1}(z/\sqrt{3})K_{\pm 2/3}\left(\zeta \right)=(z/3)\left(I_{2/3}\left(\zeta \right)-I_{-2/3}\left(\zeta \right)\right)=\frac{1}{2}(z/\sqrt{3}){\mathrm{e}}^{-\pi \mathrm{i}/6}{H}_{2/3}^{(1)}\left(\zeta {\mathrm{e}}^{\pi \mathrm{i}/2}\right)=\frac{1}{2}(z/\sqrt{3}){\mathrm{e}}^{-5\pi \mathrm{i}/6}{H}_{-2/3}^{(1)}\left(\zeta {\mathrm{e}}^{\pi \mathrm{i}/2}\right)=\frac{1}{2}(z/\sqrt{3}){\mathrm{e}}^{\pi \mathrm{i}/6}{H}_{2/3}^{(2)}\left(\zeta {\mathrm{e}}^{-\pi \mathrm{i}/2}\right)=\frac{1}{2}(z/\sqrt{3}){\mathrm{e}}^{5\pi \mathrm{i}/6}{H}_{-2/3}^{(2)}\left(\zeta {\mathrm{e}}^{-\pi \mathrm{i}/2}\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $z$: complex variable and $\zeta$: change of variable

Proof sketch:

Derivable from those given in Miller (1946, p. B17) and Olver (1997b, pp. 392–393).

A&S Ref:

10.4.16 (with opposite sign!)

Referenced by:

(9.6.22)

Permalink:

http://dlmf.nist.gov/9.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(i), §9.6 and Ch.9

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9.6.17 $$H_{1/3}^{(1)}\left(\zeta \right)={\mathrm{e}}^{-\pi \mathrm{i}/3}{H}_{-1/3}^{(1)}\left(\zeta \right)={\mathrm{e}}^{-\pi \mathrm{i}/6}\sqrt{3/z}\left(\mathrm{Ai}\left(-z\right)-\mathrm{i}\mathrm{Bi}\left(-z\right)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta (z)$: change of variable

Proof sketch:

Derivable from those given in Olver (1997b, pp. 392–393).

A&S Ref:

10.4.23 (with different sign, different form!)

Referenced by:

(9.8.11), (9.8.9)

Permalink:

http://dlmf.nist.gov/9.6.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(ii), §9.6 and Ch.9

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28: 28.4 Fourier Series

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§28.4(v) Change of Sign of $q$

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29: 28.28 Integrals, Integral Representations, and Integral Equations

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§28.28(i) Equations with Elementary Kernels

… ►where the upper or lower sign is taken according as $0\le y\le \pi$ or $\pi \le y\le 2\pi$. … ►

§28.28(ii) Integrals of Products with Bessel Functions

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§28.28(iv) Integrals of Products of Mathieu Functions of Integer Order

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§28.28(v) Compendia

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30: 19.3 Graphics

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Figure 19.3.5: $\mathrm{\Pi }({\alpha }^2,k)$ as a function of $k^2$ and ${\alpha }^2$ for , $-2\le {\alpha }^2\le 2$. …The function is unbounded as ${\alpha }^2\to 1-$, and also (with the same sign as $1-{\alpha }^2$) as $k^2\to 1-$. … Magnify 3D Help

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