principal branch (value)

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11: 25.12 Polylogarithms

… ►Other notations and names for ${\mathrm{Li}}_2\left(z\right)$ include $S_2(z)$ (Kölbig et al. (1970)), Spence function $\mathrm{Sp}(z)$ (’t Hooft and Veltman (1979)), and ${\mathrm{L}}_2(z)$ (Maximon (2003)). …

12: 4.3 Graphics

… ►Figure 4.3.2 illustrates the conformal mapping of the strip onto the whole $w$-plane cut along the negative real axis, where $w={\mathrm{e}}^z$ and $z=\ln w$ (principal value). … ►In the graphics shown in this subsection height corresponds to the absolute value of the function and color to the phase. … ►

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Figure 4.3.3: $\ln \left(x+\mathrm{i}y\right)$ (principal value). There is a branch cut along the negative real axis. Magnify 3D Help

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13: 13.2 Definitions and Basic Properties

… ►The principal branch corresponds to the principal value of $z^{-a}$ in (13.2.6), and has a cut in the $z$-plane along the interval $(-\mathrm{\infty },0]$; compare §4.2(i). …

14: 4.23 Inverse Trigonometric Functions

… ►The function ${(1-t^2)}^{1/2}$ assumes its principal value when $t\in (-1,1)$; elsewhere on the integration paths the branch is determined by continuity. … ►The principal values (or principal branches) of the inverse sine, cosine, and tangent are obtained by introducing cuts in the $z$-plane as indicated in Figures 4.23.1(i) and 4.23.1(ii), and requiring the integration paths in (4.23.1)–(4.23.3) not to cross these cuts. … ►

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

►

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

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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 general value.)

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

…

15: 6.4 Analytic Continuation

… ►Analytic continuation of the principal value of $E_1\left(z\right)$ yields a multi-valued function with branch points at $z=0$ and $z=\mathrm{\infty }$. … ►Unless indicated otherwise, in the rest of this chapter and elsewhere in the DLMF the functions $E_1\left(z\right)$, $\mathrm{Ci}\left(z\right)$, $\mathrm{Chi}\left(z\right)$, $\mathrm{f}\left(z\right)$, and $\mathrm{g}\left(z\right)$ assume their principal values, that is, the branches that are real on the positive real axis and two-valued on the negative real axis. …

16: 13.14 Definitions and Basic Properties

… ► …

17: 6.2 Definitions and Interrelations

… ►

6.2.8 $$\mathrm{li}\left(x\right)={⨍}_0^x\frac{dt}{\ln t}=\mathrm{Ei}\left(\ln x\right),$$ $x>1$.

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Defines:

$\mathrm{li}\left(x\right)$: logarithmic integral

Symbols:

$dx$: differential of $x$ , $\mathrm{Ei}\left(x\right)$: exponential integral, $\ln z$: principal branch of logarithm function, ${⨍}_a^b$: Cauchy principal value and $x$: real variable

A&S Ref:

5.1.3

Referenced by:

§27.12, §6.12(i)

Permalink:

http://dlmf.nist.gov/6.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(i), §6.2 and Ch.6

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18: 8.19 Generalized Exponential Integral

… ►When the path of integration excludes the origin and does not cross the negative real axis (8.19.2) defines the principal value of $E_p\left(z\right)$, and unless indicated otherwise in the DLMF principal values are assumed. … ►In Figures 8.19.2–8.19.5, height corresponds to the absolute value of the function and color to the phase. … ►

§8.19(iii) Special Values

… ►Unless $p$ is a nonpositive integer, $E_p\left(z\right)$ has a branch point at $z=0$. For $z\ne 0$ each branch of $E_p\left(z\right)$ is an entire function of $p$. …

19: Mathematical Introduction

… ►These include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§1.10(vi), 4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§1.17); numerically satisfactory solutions of differential and difference equations (§§2.7(iv), 2.9(i)); and numerical analysis for complex variables (Chapter 3). …

20: 10.41 Asymptotic Expansions for Large Order

… ►where the branches assume their principal values. …