# principal branch (value)

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## 1—10 of 98 matching pages

##### 1: 4.1 Special Notation

##### 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 $(-\mathrm{\infty},0]$. … ► … ►##### 3: 4.2 Definitions

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

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►This result is also valid when ${z}^{a}$ has its principal value, provided that the branch of $\mathrm{Ln}w$ satisfies
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##### 4: 16.2 Definition and Analytic Properties

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►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 ${}_{q+1}F_{q}(\mathbf{a};\mathbf{b};z)$; 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

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►When $z$ is complex ${P}_{\nu}^{\pm \mu}\left(z\right)$, ${Q}_{\nu}^{\mu}\left(z\right)$, and ${\mathit{Q}}_{\nu}^{\mu}\left(z\right)$ 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).
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##### 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 $\u2102\setminus (-\mathrm{\infty},0]$, 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 $\u2102\setminus (-\mathrm{\infty},0]$, and two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi $. …##### 7: 15.2 Definitions and Analytical Properties

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►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 4.15.7 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 $w=\mathrm{sin}z$ and $z=\mathrm{arcsin}w$ (principal value).
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►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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##### 9: 4.37 Inverse Hyperbolic Functions

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*principal values*(or*principal branches*) of the inverse $\mathrm{sinh}$, $\mathrm{cosh}$, and $\mathrm{tanh}$ 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
$${\u2a0d}_{0}^{x}\frac{dt}{\mathrm{ln}t}=\mathrm{li}\left(x\right),$$
$x>1$.

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