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##### 1: 4.13 Lambert $W$-Function
We call the solution for which $W\left(x\right)\geq W\left(-1/e\right)$ the principal branch and denote it by $\mathrm{Wp}\left(x\right)$. … Figure 4.13.1: Branches Wp ⁡ ( x ) and Wm ⁡ ( x ) of the Lambert W -function. … Magnify
##### 3: 1.10 Functions of a Complex Variable
If $D=\mathbb{C}\setminus(-\infty,0]$ and $z=re^{i\theta}$, then one branch is $\sqrt{r}e^{i\theta/2}$, the other branch is $-\sqrt{r}e^{i\theta/2}$, with $-\pi<\theta<\pi$ in both cases. Similarly if $D=\mathbb{C}\setminus[0,\infty)$, then one branch is $\sqrt{r}e^{i\theta/2}$, the other branch is $-\sqrt{r}e^{i\theta/2}$, with $0<\theta<2\pi$ in both cases. … (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 $F((z_{0}-a)e^{2k\pi i}+a)$. …
##### 4: 15.2 Definitions and Analytical Properties
again with analytic continuation for other values of $z$, and with the principal branch defined in a similar way. … The same is true of other branches, provided that $z=0$, $1$, and $\infty$ are excluded. …
##### 5: 4.37 Inverse Hyperbolic Functions
$\operatorname{Arcsinh}z$ and $\operatorname{Arccsch}z$ have branch points at $z=\pm i$; the other four functions have branch points at $z=\pm 1$. …
##### 6: 10.21 Zeros
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). …
##### 7: 31.11 Expansions in Series of Hypergeometric Functions
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 $\mathcal{E}$. …
##### 8: 14.24 Analytic Continuation
For fixed $z$, other than $\pm 1$ or $\infty$, each branch of $P^{-\mu}_{\nu}\left(z\right)$ and $\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)$ is an entire function of each parameter $\nu$ and $\mu$. …
##### 9: 4.23 Inverse Trigonometric Functions
$\operatorname{Arctan}z$ and $\operatorname{Arccot}z$ have branch points at $z=\pm\mathrm{i}$; the other four functions have branch points at $z=\pm 1$. …
##### 10: 28.7 Analytic Continuation of Eigenvalues
The branch points are called the exceptional values, and the other points normal values. …