general values

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1: 5.21 Methods of Computation

… ►An effective way of computing $\mathrm{\Gamma }\left(z\right)$ in the right half-plane is backward recurrence, beginning with a value generated from the asymptotic expansion (5.11.3). …

2: 8.24 Physical Applications

… ►With more general values of $p$, $E_p\left(x\right)$ supplies fundamental auxiliary functions that are used in the computation of molecular electronic integrals in quantum chemistry (Harris (2002), Shavitt (1963)), and also wave acoustics of overlapping sound beams (Ding (2000)).

3: 8.21 Generalized Sine and Cosine Integrals

… ►

§8.21(i) Definitions: General Values

►With $\gamma$ and $\mathrm{\Gamma }$ denoting here the general values of the incomplete gamma functions (§8.2(i)), we define … ►

§8.21(ii) Definitions: Principal Values

►When $\mathrm{ph}z=0$ (and when $a\ne -1,-3,-5,\mathrm{\dots }$, in the case of $\mathrm{Si}(a,z)$, or $a\ne 0,-2,-4,\mathrm{\dots }$, in the case of $\mathrm{Ci}(a,z)$) the principal values of $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$ are defined by (8.21.1) and (8.21.2) with the incomplete gamma functions assuming their principal values (§8.2(i)). … ►

§8.21(v) Special Values

…

4: 25.18 Methods of Computation

… ►The principal tools for computing $\zeta \left(s\right)$ are the expansion (25.2.9) for general values of $s$, and the Riemann–Siegel formula (25.10.3) (extended to higher terms) for $\zeta \left(\frac{1}{2}+\mathrm{i}t\right)$. …

5: 6.4 Analytic Continuation

… ►The general value of $E_1\left(z\right)$ is given by … ►The general values of the other functions are defined in a similar manner, and …

6: 4.2 Definitions

… ►

§4.2(i) The Logarithm

… ►The general value of the phase is given by … ►

Powers with General Bases

… ►but the general value of ${\mathrm{e}}^z$ is … ►If $z^a$ has its general value, with $a\ne 0$, and if $w\ne 0$, then …

7: 8.2 Definitions and Basic Properties

… ►The general values of the incomplete gamma functions $\gamma (a,z)$ and $\mathrm{\Gamma }(a,z)$ are defined by … ►In this subsection the functions $\gamma$ and $\mathrm{\Gamma }$ have their general values. …

8: 4.37 Inverse Hyperbolic Functions

… ►

§4.37(i) General Definitions

►The general values of the inverse hyperbolic functions are defined by …

9: 4.23 Inverse Trigonometric Functions

… ►

§4.23(i) General Definitions

►The general values of the inverse trigonometric functions are defined by … ►

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 generalvalue.)

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

►

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 generalvalue.)

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

►

4.23.36 $$\arctan z=\frac{1}{2}\arctan \left(\frac{2x}{1-x^2-y^2}\right)+\frac{1}{4}\mathrm{i}\ln \left(\frac{x^2+{(y+1)}^2}{x^2+{(y-1)}^2}\right),$$

ⓘ

Symbols:

$\mathrm{i}$: imaginary unit, $\arctan z$: arctangent function, $\ln z$: principal branch of logarithm function, $x$: real variable, $y$: real variable and $z$: complex variable

A&S Ref:

4.4.39 (with the generalvalue.)

Referenced by:

(4.23.34), (4.23.35), §4.23(vi)

Permalink:

http://dlmf.nist.gov/4.23.E36

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(vi), §4.23 and Ch.4

…

10: 8.19 Generalized Exponential Integral

… ►

8.19.1 $$E_p\left(z\right)=z^{p-1}\mathrm{\Gamma }(1-p,z).$$

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Defines:

$E_p\left(z\right)$: generalized exponential integral

Symbols:

$\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable and $p$: parameter

A&S Ref:

5.1.45 6.5.9 (Definition extended to generalvalues of $p$.)

Referenced by:

§8.19(i), §8.19(iii), §8.19(iv), §8.19(v), §8.19(vi), §8.19(vii)

Permalink:

http://dlmf.nist.gov/8.19.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(i), §8.19 and Ch.8

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8.19.2 $$E_p\left(z\right)=z^{p-1}{\int }_z^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t}}{t^p}dt.$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\int$: integral, $z$: complex variable and $p$: parameter

Keywords:

Laplace transform, Mellin transform

Referenced by:

§8.19(i), §8.19(viii)

Permalink:

http://dlmf.nist.gov/8.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(i), §8.19 and Ch.8

… ►

§8.19(iii) Special Values

… ►

8.19.6 $$E_p\left(0\right)=\frac{1}{p-1},$$ $\mathrm{\Re }p>1$,

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Symbols:

$E_p\left(z\right)$: generalized exponential integral, $\mathrm{\Re }$: real part and $p$: parameter

A&S Ref:

5.1.23 (Extended to generalvalues of $p$.)

Permalink:

http://dlmf.nist.gov/8.19.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(iii), §8.19 and Ch.8

… ►

8.19.13 $$\frac{d}{dz}{E}_p\left(z\right)=-E_{p-1}\left(z\right),$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $E_p\left(z\right)$: generalized exponential integral, $z$: complex variable and $p$: parameter

A&S Ref:

5.1.36 (Extended to generalvalues of $p$.)

Referenced by:

§8.19(x)

Permalink:

http://dlmf.nist.gov/8.19.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(v), §8.19 and Ch.8

…