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## 1—10 of 94 matching pages

##### 1: 4.13 Lambert $W$-Function

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►We call the increasing solution for which $W\left(z\right)\ge W\left(-{\mathrm{e}}^{-1}\right)=-1$ the

*principal branch*and denote it by ${W}_{0}\left(z\right)$. … ► … ►Other solutions of (4.13.1) are other branches of $W\left(z\right)$. …The other branches ${W}_{k}\left(z\right)$ are single-valued analytic functions on $\u2102\setminus (-\mathrm{\infty},0]$, have a logarithmic branch point at $z=0$, and, in the case $k=\pm 1$, have a square root branch point at $z=-{\mathrm{e}}^{-1}\mp 0\mathrm{i}$ respectively. …##### 2: 4.24 Inverse Trigonometric Functions: Further Properties

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##### 3: 1.10 Functions of a Complex Variable

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►If $D=\u2102\setminus (-\mathrm{\infty},0]$ and $z=r{\mathrm{e}}^{\mathrm{i}\theta}$, then one branch is $\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, the other branch is $-\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, with $$ in both cases.
Similarly if $D=\u2102\setminus [0,\mathrm{\infty})$, then one branch is $\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, the other branch is $-\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, with $$ in both cases.
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►(b) By specifying the value of $F(z)$ at a point ${z}_{0}$ (not a branch point), and requiring $F(z)$ to be continuous on any path that begins at ${z}_{0}$ and does not pass through any branch points or other singularities of $F(z)$.
►If the path circles a branch point at $z=a$, $k$ times in the positive sense, and returns to ${z}_{0}$ without encircling any other branch point, then its value is denoted conventionally as $F(({z}_{0}-a){\mathrm{e}}^{2k\pi \mathrm{i}}+a)$.
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##### 4: 15.2 Definitions and Analytical Properties

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►again with analytic continuation for other values of $z$, and with the principal branch defined in a similar way.
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►The same is true of other branches, provided that $z=0$, $1$, and $\mathrm{\infty}$ are excluded.
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##### 5: 4.37 Inverse Hyperbolic Functions

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$\mathrm{Arcsinh}z$ and $\mathrm{Arccsch}z$ have branch points at $z=\pm \mathrm{i}$; the other four functions have branch points at $z=\pm 1$.
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##### 6: 10.21 Zeros

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►For further information, including uniform asymptotic expansions, extensions to other branches of the functions and their derivatives, and extensions to half-integer values of the order, see Olver (1954).
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##### 7: 31.11 Expansions in Series of Hypergeometric Functions

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►The expansion (31.11.1) for a Heun function that is associated with any branch of (31.11.2)—other than a multiple of the right-hand side of (31.11.12)—is convergent inside the ellipse $\mathcal{E}$.
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##### 8: 14.24 Analytic Continuation

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►For fixed $z$, other than $\pm 1$ or $\mathrm{\infty}$, each branch of ${P}_{\nu}^{-\mu}\left(z\right)$ and ${\bm{Q}}_{\nu}^{\mu}\left(z\right)$ is an entire function of each parameter $\nu $ and $\mu $.
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##### 9: 4.23 Inverse Trigonometric Functions

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$\mathrm{Arctan}z$ and $\mathrm{Arccot}z$ have branch points at $z=\pm \mathrm{i}$; the other four functions have branch points at $z=\pm 1$.
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