# branch cuts

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##### 1: 4.2 Definitions
Most texts extend the definition of the principal value to include the branch cut
##### 4: 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. …
##### 7: 4.23 Inverse Trigonometric Functions
The principal values (or principal branches) of the inverse sine, cosine, and tangent are obtained by introducing cuts in the $z$-plane as indicated in Figures 4.23.1(i) and 4.23.1(ii), and requiring the integration paths in (4.23.1)–(4.23.3) not to cross these cuts. …
##### 8: 4.13 Lambert $W$-Function
where $t\geq 0$ for $W_{0}$, $t\leq 0$ for $W_{\pm 1}$ on the relevant branch cuts, …
##### 9: 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 $F\left(a,b;c;z\right)$. … The difference between the principal branches on the two sides of the branch cut4.2(i)) is given by …
##### 10: 3.10 Continued Fractions
For example, by converting the Maclaurin expansion of $\operatorname{arctan}z$ (4.24.3), we obtain a continued fraction with the same region of convergence ($\left|z\right|\leq 1$, $z\neq\pm\mathrm{i}$), whereas the continued fraction (4.25.4) converges for all $z\in\mathbb{C}$ except on the branch cuts from $i$ to $i\infty$ and $-i$ to $-i\infty$. …