principal branch (value)

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1: 4.1 Special Notation

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$k,m,n$	integers.
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2: 10.2 Definitions

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§10.2(ii) Standard Solutions

… ►The principal branch of $J_{\nu }\left(z\right)$ corresponds to the principal value of ${(\frac{1}{2}z)}^{\nu }$ (§4.2(iv)) and is analytic in the $z$-plane cut along the interval $(-\mathrm{\infty },0]$. … ►The principal branches correspond to principal values of the square roots in (10.2.5) and (10.2.6), again with a cut in the $z$-plane along the interval $(-\mathrm{\infty },0]$. … ► … ►

Table 10.2.1: Numerically satisfactory pairs of solutions of Bessel’s equation.

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Pair	Interval or Region
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3: 4.2 Definitions

… ►This is a multivalued function of $z$ with branch point at $z=0$. ►The principal value, or principal branch, is defined by … ►Most texts extend the definition of the principal value to include the branch cut … ►

4.2.18 $$\ln 10=\mathrm{2.30258\; 50929\; 94045\; 68401}\mathrm{\dots }.$$

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Symbols:

$\ln z$: principal branch of logarithm function

A&S Ref:

4.1.23 (with 10D value.)

Notes:

For more digits see OEIS Sequence A002392; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/4.2.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §4.2(ii), §4.2 and Ch.4

… ►This result is also valid when $z^a$ has its principal value, provided that the branch of $\mathrm{Ln}w$ satisfies …

4: 16.2 Definition and Analytic Properties

… ►The branch obtained by introducing a cut from $1$ to $+\mathrm{\infty }$ on the real axis, that is, the branch in the sector $|\mathrm{ph}\left(1-z\right)|\le \pi$, is the principal branch (or principal value) of ${}_{q+1}{}^{}F_q^{}(𝐚;𝐛;z)$; compare §4.2(i). …Unless indicated otherwise it is assumed that in the DLMF generalized hypergeometric functions assume their principal values. …

5: 14.21 Definitions and Basic Properties

… ►When $z$ is complex $P_{\nu }^{\pm \mu }\left(z\right)$, $Q_{\nu }^{\mu }\left(z\right)$, and ${𝑸}_{\nu }^{\mu }\left(z\right)$ are defined by (14.3.6)–(14.3.10) with $x$ replaced by $z$: the principal branches are obtained by taking the principal values of all the multivalued functions appearing in these representations when $z\in (1,\mathrm{\infty })$, and by continuity elsewhere in the $z$-plane with a cut along the interval $(-\mathrm{\infty },1]$; compare §4.2(i). … …

6: 10.25 Definitions

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§10.25(ii) Standard Solutions

… ►In particular, the principal branch of $I_{\nu }\left(z\right)$ is defined in a similar way: it corresponds to the principal value of ${(\frac{1}{2}z)}^{\nu }$, is analytic in $ℂ\setminus (-\mathrm{\infty },0]$, and two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi$. … ►The principal branch corresponds to the principal value of the square root in (10.25.3), is analytic in $ℂ\setminus (-\mathrm{\infty },0]$, and two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi$. …

7: 15.2 Definitions and Analytical Properties

… ►The branch obtained by introducing a cut from $1$ to $+\mathrm{\infty }$ on the real $z$-axis, that is, the branch in the sector $|\mathrm{ph}\left(1-z\right)|\le \pi$, is the principal branch (or principal value) of $F(a,b;c;z)$. … ►again with analytic continuation for other values of $z$, and with the principal branch defined in a similar way. …

8: 4.15 Graphics

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Figure 4.15.2: $\mathrm{Arcsin}x$ and $\mathrm{Arccos}x$. Principal values are shown with thickened lines. Magnify

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Figure 4.15.4: $\arctan x$ and $\mathrm{arccot}x$. Only principal values are shown. … Magnify

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Figure 4.15.6: $\mathrm{arccsc}x$ and $\mathrm{arcsec}x$. Only principal values are shown. … Magnify

… ►Figure 4.15.7 illustrates the conformal mapping of the strip onto the whole $w$-plane cut along the real axis from $-\mathrm{\infty }$ to $-1$ and $1$ to $\mathrm{\infty }$, where $w=\sin z$ and $z=\arcsin w$ (principal value). … ►In the graphics shown in this subsection height corresponds to the absolute value of the function and color to the phase. …

9: 4.37 Inverse Hyperbolic Functions

… ►The principal values (or principal branches) of the inverse $\sinh$, $\cosh$, and $\tanh$ are obtained by introducing cuts in the $z$-plane as indicated in Figure 4.37.1(i)-(iii), and requiring the integration paths in (4.37.1)–(4.37.3) not to cross these cuts. …

10: 4.10 Integrals

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4.10.7 $${⨍}_0^x\frac{dt}{\ln t}=\mathrm{li}\left(x\right),$$ $x>1$.

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Symbols:

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

A&S Ref:

4.1.57

Referenced by:

§4.10(i)

Permalink:

http://dlmf.nist.gov/4.10.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §4.10(i), §4.10 and Ch.4

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