principal branch (value)

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1: 4.1 Special Notation
 $k,m,n$ integers. …
2: 10.2 Definitions
§10.2(ii) Standard Solutions
The principal branch of $J_{\nu}\left(z\right)$ corresponds to the principal value of $(\tfrac{1}{2}z)^{\nu}$4.2(iv)) and is analytic in the $z$-plane cut along the interval $(-\infty,0]$. … 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 $(-\infty,0]$. …
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 $\ln 10=2.30258\ 50929\ 94045\ 68401\dots.$
This result is also valid when $z^{a}$ has its principal value, provided that the branch of $\operatorname{Ln}w$ satisfies …
4: 16.2 Definition and Analytic Properties
The branch obtained by introducing a cut from $1$ to $+\infty$ on the real axis, that is, the branch in the sector $|\operatorname{ph}\left(1-z\right)|\leq\pi$, is the principal branch (or principal value) of ${{}_{q+1}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)$; 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 $P^{\pm\mu}_{\nu}\left(z\right)$, $Q^{\mu}_{\nu}\left(z\right)$, and $\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)$ 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 $z\in(1,\infty)$, and by continuity elsewhere in the $z$-plane with a cut along the interval $(-\infty,1]$; compare §4.2(i). … …
6: 10.25 Definitions
§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 $(\tfrac{1}{2}z)^{\nu}$, is analytic in $\mathbb{C}\setminus(-\infty,0]$, and two-valued and discontinuous on the cut $\operatorname{ph}z=\pm\pi$. … The principal branch corresponds to the principal value of the square root in (10.25.3), is analytic in $\mathbb{C}\setminus(-\infty,0]$, and two-valued and discontinuous on the cut $\operatorname{ph}z=\pm\pi$. …
7: 15.2 Definitions and Analytical Properties
The branch obtained by introducing a cut from $1$ to $+\infty$ on the real $z$-axis, that is, the branch in the sector $|\operatorname{ph}\left(1-z\right)|\leq\pi$, is the principal branch (or principal value) of $F\left(a,b;c;z\right)$. … again with analytic continuation for other values of $z$, and with the principal branch defined in a similar way. …
8: 4.15 Graphics
Figure 4.15.7 illustrates the conformal mapping of the strip $-\tfrac{1}{2}\pi<\Re z<\tfrac{1}{2}\pi$ onto the whole $w$-plane cut along the real axis from $-\infty$ to $-1$ and $1$ to $\infty$, where $w=\sin z$ and $z=\operatorname{arcsin}w$ (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. …