sign function

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31—40 of 97 matching pages

31: 23.5 Special Lattices

… ►

§23.5(i) Real-Valued Functions

… ►Also,

e_{2}

and

g_{3}

have opposite signs unless

\omega_{3}=i\omega_{1}

, in which event both are zero. … ►

§23.5(iii) Lemniscatic Lattice

… ►

§23.5(iv) Rhombic Lattice

… ►

e_{1}

and

g_{3}

have the same sign unless

2\omega_{3}=(1+i)\omega_{1}

when both are zero: the pseudo-lemniscatic case. …

32: 19.14 Reduction of General Elliptic Integrals

… ►

19.14.3

\int_{0}^{x}\frac{\,\mathrm{d}t}{\sqrt{1+t^{4}}}=\frac{\operatorname{sign}% \left(x\right)}{2}F\left(\phi,k\right),

\cos\phi=\dfrac{1-x^{2}}{1+x^{2}}

k^{2}=\dfrac{1}{2}

ⓘ

Symbols:: $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\int$ : integral, $\operatorname{sign}\NVar{x}$ : sign of, $\phi$ : 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

…

33: 12.10 Uniform Asymptotic Expansions for Large Parameter

… ►With the upper sign in (12.10.2), expansions can be constructed for large

\mu

in terms of elementary functions that are uniform for

t\in(-\infty,\infty)

(§2.8(ii)). …

34: 11.6 Asymptotic Expansions

… ►

§11.6(i) Large $|z|$ , Fixed $\nu$

… ►If

\nu

is real,

z

is positive, and

m+\tfrac{1}{2}-\nu\geq 0

, then

R_{m}(z)

is of the same sign and numerically less than the first neglected term. … ►

§11.6(ii) Large $|\nu|$ , Fixed $z$

… ► … ►Here …

35: 15.9 Relations to Other Functions

… ►

§15.9(ii) Jacobi Function

… ►

§15.9(iii) Gegenbauer Function

… ►

§15.9(iv) Associated Legendre Functions; Ferrers Functions

… ►where the sign in the exponential is

\pm

according as

\Im z\gtrless 0

. …where the sign in the exponential is

\pm

according as

\Im z\gtrless 0

. …

36: 9.7 Asymptotic Expansions

§9.7 Asymptotic Expansions

… ►

§9.7(iii) Error Bounds for Real Variables

►In (9.7.5) and (9.7.6) the

n

th error term, that is, the error on truncating the expansion at

n

terms, is bounded in magnitude by the first neglected term and has the same sign, provided that the following term is of opposite sign, that is, if

n\geq 0

for (9.7.5) and

n\geq 1

for (9.7.6). … ►In (9.7.9)–(9.7.12) the

n

th error term in each infinite series is bounded in magnitude by the first neglected term and has the same sign, provided that the following term in the series is of opposite sign. … ►

►The first section in each of the special function chapters (Chapters 5–36) lists notation that has been adopted for the functions in that chapter. … ►Similarly in the case of confluent hypergeometric functions (§13.2(i)). … ►Special functions with a complex variable are depicted as colored 3D surfaces in a similar way to functions of two real variables, but with the vertical height corresponding to the modulus (absolute value) of the function. … ►All of the special function chapters contain sections that describe available methods for computing the main functions in the chapter, and most also provide references to numerical tables of, and approximations for, these functions. …

38: Errata

… ►

Additions

Equations: (5.9.2_5), (5.9.10_1), (5.9.10_2), (5.9.11_1), (5.9.11_2), the definition of the scaled gamma function $\Gamma^{*}\left(z\right)$ was inserted after the first equals sign in (5.11.3), post equality added in (7.17.2) which gives “ $=\sum_{m=0}^{\infty}a_{m}t^{2m+1}$ ”, (7.17.2_5), (31.11.3_1), (31.11.3_2) with some explanatory text.

… ►

Table 22.4.3

Originally a minus sign was missing in the entries for $\operatorname{cd}u$ and $\operatorname{dc}u$ in the second column (headed $z+K+iK^{\prime}$ ). The correct entries are $-k^{-1}\operatorname{ns}z$ and $-k\operatorname{sn}z$ . Note: These entries appear online but not in the published print edition. More specifically, Table 22.4.3 in the published print edition is restricted to the three Jacobian elliptic functions $\operatorname{sn},\operatorname{cn},\operatorname{dn}$ , whereas Table 22.4.3 covers all 12 Jacobian elliptic functions.

	$u$
	$z+K$	$z+K+\mathrm{i}K^{\prime}$	$z+\mathrm{i}K^{\prime}$	$z+2K$	$z+2K+2\mathrm{i}K^{\prime}$	$z+2\mathrm{i}K^{\prime}$
$\vdots$
$\operatorname{cd}u$	$-\operatorname{sn}z$	$-k^{-1}\operatorname{ns}z$	$k^{-1}\operatorname{dc}z$	$-\operatorname{cd}z$	$-\operatorname{cd}z$	$\operatorname{cd}z$
$\vdots$
$\operatorname{dc}u$	$-\operatorname{ns}z$	$-k\operatorname{sn}z$	$k\operatorname{cd}z$	$-\operatorname{dc}z$	$-\operatorname{dc}z$	$\operatorname{dc}z$
$\vdots$

Reported 2014-02-28 by Svante Janson.

…

39: 13.8 Asymptotic Approximations for Large Parameters

… ►

§13.8(ii) Large $b$ and $z$ , Fixed $a$ and $b/z$

►Let

\lambda=z/b>0

and

\zeta=\sqrt{2(\lambda-1-\ln\lambda)}

with

\operatorname{sign}\left(\zeta\right)=\operatorname{sign}\left(\lambda-1\right)

. … ►

§13.8(iii) Large $a$

… ► … ►

§13.8(iv) Large $a$ and $b$

…

40: 31.3 Basic Solutions

… ►

31.3.10

z^{-\alpha}\mathit{H\!\ell}\left(\frac{1}{a},\frac{q}{a}-\alpha\left(\beta-% \epsilon\right)-\frac{\alpha}{a}\left(\beta-\delta\right);\alpha,\alpha-\gamma% +1,\alpha-\beta+1,\delta;\frac{1}{z}\right),

ⓘ

Symbols:: $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Referenced by:: §31.11(iii), §31.11(iii), Erratum (V1.1.7) for Equations (31.3.10), (31.3.11), Erratum (V1.1.7) for Subsection 31.11(iii)
Permalink:: http://dlmf.nist.gov/31.3.E10
Encodings:: pMML, png, TeX
Correction (effective with 1.1.7):: The second entry in the $\mathit{H\!\ell}$ has been corrected with an extra minus sign.
See also:: Annotations for §31.3(ii), §31.3 and Ch.31

►

31.3.11

z^{-\beta}\mathit{H\!\ell}\left(\frac{1}{a},\frac{q}{a}-\beta\left(\alpha-% \epsilon\right)-\frac{\beta}{a}\left(\alpha-\delta\right);\beta,\beta-\gamma+1% ,\beta-\alpha+1,\delta;\frac{1}{z}\right).

ⓘ

Symbols:: $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Referenced by:: §31.3(iii), Erratum (V1.1.7) for Equations (31.3.10), (31.3.11)
Permalink:: http://dlmf.nist.gov/31.3.E11
Encodings:: pMML, png, TeX
Correction (effective with 1.1.7):: The second entry in the $\mathit{H\!\ell}$ has been corrected with an extra minus sign.
See also:: Annotations for §31.3(ii), §31.3 and Ch.31

…

sign function

31—40 of 97 matching pages

31: 23.5 Special Lattices

§23.5(i) Real-Valued Functions

§23.5(iii) Lemniscatic Lattice

§23.5(iv) Rhombic Lattice

32: 19.14 Reduction of General Elliptic Integrals

33: 12.10 Uniform Asymptotic Expansions for Large Parameter

34: 11.6 Asymptotic Expansions

§11.6(i) Large $|z|$ , Fixed $\nu$

§11.6(ii) Large $|\nu|$ , Fixed $z$

35: 15.9 Relations to Other Functions

§15.9(ii) Jacobi Function

§15.9(iii) Gegenbauer Function

§15.9(iv) Associated Legendre Functions; Ferrers Functions

36: 9.7 Asymptotic Expansions

§9.7 Asymptotic Expansions

§9.7(iii) Error Bounds for Real Variables

37: Mathematical Introduction

Notation for the Special Functions

38: Errata

39: 13.8 Asymptotic Approximations for Large Parameters

§13.8(ii) Large $b$ and $z$ , Fixed $a$ and $b/z$

§13.8(iii) Large $a$

§13.8(iv) Large $a$ and $b$

40: 31.3 Basic Solutions

sign function

31—40 of 97 matching pages

31: 23.5 Special Lattices

§23.5(i) Real-Valued Functions

§23.5(iii) Lemniscatic Lattice

§23.5(iv) Rhombic Lattice

32: 19.14 Reduction of General Elliptic Integrals

33: 12.10 Uniform Asymptotic Expansions for Large Parameter

34: 11.6 Asymptotic Expansions

§11.6(i) Large | z | , Fixed ν

§11.6(ii) Large | ν | , Fixed z

35: 15.9 Relations to Other Functions

§15.9(ii) Jacobi Function

§15.9(iii) Gegenbauer Function

§15.9(iv) Associated Legendre Functions; Ferrers Functions

36: 9.7 Asymptotic Expansions

§9.7 Asymptotic Expansions

§9.7(iii) Error Bounds for Real Variables

37: Mathematical Introduction

Notation for the Special Functions

38: Errata

39: 13.8 Asymptotic Approximations for Large Parameters

§13.8(ii) Large b and z , Fixed a and b / z

§13.8(iii) Large a

§13.8(iv) Large a and b

40: 31.3 Basic Solutions

§11.6(i) Large $|z|$ , Fixed $\nu$

§11.6(ii) Large $|\nu|$ , Fixed $z$

§13.8(ii) Large $b$ and $z$ , Fixed $a$ and $b/z$

§13.8(iii) Large $a$

§13.8(iv) Large $a$ and $b$