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1: 4.13 Lambert $W$-Function

… ►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 $W_0\left(z\right)$. … ►

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Figure 4.13.1: Branches $W_0\left(x\right)$, $W_{\pm 1}\left(x\mp 0\mathrm{i}\right)$ of the Lambert $W$-function. Magnify

… ►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 $ℂ\setminus (-\mathrm{\infty },0]$, 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

… ►If $D=ℂ\setminus (-\mathrm{\infty },0]$ 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 $D=ℂ\setminus [0,\mathrm{\infty })$, 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. … ►(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 $F((z_0-a){\mathrm{e}}^{2k\pi \mathrm{i}}+a)$. …

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 $\mathrm{\infty }$ are excluded. …

5: 4.37 Inverse Hyperbolic Functions

… ► $\mathrm{Arcsinh}z$ and $\mathrm{Arccsch}z$ have branch points at $z=\pm \mathrm{i}$; the other four functions have branch points at $z=\pm 1$. …

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 $ℰ$. …

8: 14.24 Analytic Continuation

… ►For fixed $z$, other than $\pm 1$ or $\mathrm{\infty }$, each branch of $P_{\nu }^{-\mu }\left(z\right)$ and ${𝑸}_{\nu }^{\mu }\left(z\right)$ is an entire function of each parameter $\nu$ and $\mu$. …

9: 4.23 Inverse Trigonometric Functions

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

10: 28.7 Analytic Continuation of Eigenvalues

… ►The branch points are called the exceptional values, and the other points normal values. …