# of multivalued function

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##### 1: 4.2 Definitions
The general logarithm function $\operatorname{Ln}z$ is defined by …This is a multivalued function of $z$ with branch point at $z=0$. … Most texts extend the definition of the principal value to include the branch cut In all other cases, $z^{a}$ is a multivalued function with branch point at $z=0$. …
##### 2: 1.10 Functions of a Complex Variable
###### §1.10(vi) MultivaluedFunctions
Let $F(z)$ be a multivalued function and $D$ be a domain. … Branches can be constructed in two ways: …
##### 3: 14.25 Integral Representations
where the multivalued functions have their principal values when $1 and are continuous in $\mathbb{C}\setminus(-\infty,1]$. …
##### 4: 4.37 Inverse Hyperbolic Functions
4.37.4 $\operatorname{Arccsch}z=\operatorname{Arcsinh}\left(1/z\right),$
4.37.5 $\operatorname{Arcsech}z=\operatorname{Arccosh}\left(1/z\right),$
4.37.6 $\operatorname{Arccoth}z=\operatorname{Arctanh}\left(1/z\right).$
Each of the six functions is a multivalued function of $z$. $\operatorname{Arcsinh}z$ and $\operatorname{Arccsch}z$ have branch points at $z=\pm i$; the other four functions have branch points at $z=\pm 1$. …
##### 5: 4.23 Inverse Trigonometric Functions
4.23.4 $\operatorname{Arccsc}z=\operatorname{Arcsin}\left(1/z\right),$
4.23.5 $\operatorname{Arcsec}z=\operatorname{Arccos}\left(1/z\right),$
4.23.6 $\operatorname{Arccot}z=\operatorname{Arctan}\left(1/z\right).$
Each of the six functions is a multivalued function of $z$. …
##### 6: 14.1 Special Notation
Multivalued functions take their principal values (§4.2(i)) unless indicated otherwise. …
##### 7: 14.21 Definitions and Basic Properties
When $z$ is complex $P^{\pm\mu}_{\nu}\left(z\right)$, $Q^{\mu}_{\nu}\left(z\right)$, and $\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)$ 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 $z\in(1,\infty)$, and by continuity elsewhere in the $z$-plane with a cut along the interval $(-\infty,1]$; compare §4.2(i). …
##### 8: 4.8 Identities
4.8.1 $\operatorname{Ln}\left(z_{1}z_{2}\right)=\operatorname{Ln}z_{1}+\operatorname{% Ln}z_{2}.$
4.8.3 $\operatorname{Ln}\frac{z_{1}}{z_{2}}=\operatorname{Ln}z_{1}-\operatorname{Ln}z% _{2},$
4.8.5 $\operatorname{Ln}\left(z^{n}\right)=n\operatorname{Ln}z,$ $n\in\mathbb{Z}$,
4.8.8 $\operatorname{Ln}\left(\exp z\right)=z+2k\pi\mathrm{i},$ $k\in\mathbb{Z}$,
##### 9: Mathematical Introduction
These include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§1.10(vi), 4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§1.17); numerically satisfactory solutions of differential and difference equations (§§2.7(iv), 2.9(i)); and numerical analysis for complex variables (Chapter 3). …
 $\mathbb{C}$ complex plane (excluding infinity). … multivalued functions. More generally, $F((z_{0}-a)e^{2k\pi i}+a)$. See §1.10(vi). …