# multivalued

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##### 1: 4.1 Special Notation
The main functions treated in this chapter are the logarithm $\ln z$, $\operatorname{Ln}z$; the exponential $\exp z$, $e^{z}$; the circular trigonometric (or just trigonometric) functions $\sin z$, $\cos z$, $\tan z$, $\csc z$, $\sec z$, $\cot z$; the inverse trigonometric functions $\operatorname{arcsin}z$, $\operatorname{Arcsin}z$, etc. ; the hyperbolic trigonometric (or just hyperbolic) functions $\sinh z$, $\cosh z$, $\tanh z$, $\operatorname{csch}z$, $\operatorname{sech}z$, $\coth z$; the inverse hyperbolic functions $\operatorname{arcsinh}z$, $\operatorname{Arcsinh}z$, etc. Sometimes in the literature the meanings of $\ln$ and $\operatorname{Ln}$ are interchanged; similarly for $\operatorname{arcsin}z$ and $\operatorname{Arcsin}z$, etc. …${\sin}^{-1}z$ for $\operatorname{arcsin}z$ and $\mathrm{Sin}^{-1}\;z$ for $\operatorname{Arcsin}z$.
##### 2: 4.24 Inverse Trigonometric Functions: Further Properties
4.24.13 $\operatorname{Arcsin}u\pm\operatorname{Arcsin}v=\operatorname{Arcsin}\left(u(1% -v^{2})^{1/2}\pm v(1-u^{2})^{1/2}\right),$
4.24.14 $\operatorname{Arccos}u\pm\operatorname{Arccos}v=\operatorname{Arccos}\left(uv% \mp((1-u^{2})(1-v^{2}))^{1/2}\right),$
4.24.15 $\operatorname{Arctan}u\pm\operatorname{Arctan}v=\operatorname{Arctan}\left(% \frac{u\pm v}{1\mp uv}\right),$
4.24.16 $\operatorname{Arcsin}u\pm\operatorname{Arccos}v=\operatorname{Arcsin}\left(uv% \pm((1-u^{2})(1-v^{2}))^{1/2}\right)=\operatorname{Arccos}\left(v(1-u^{2})^{1/% 2}\mp u(1-v^{2})^{1/2}\right),$
4.24.17 $\operatorname{Arctan}u\pm\operatorname{Arccot}v=\operatorname{Arctan}\left(% \frac{uv\pm 1}{v\mp u}\right)=\operatorname{Arccot}\left(\frac{v\mp u}{uv\pm 1% }\right).$
##### 3: 4.8 Identities
4.8.1 $\operatorname{Ln}\left(z_{1}z_{2}\right)=\operatorname{Ln}z_{1}+\operatorname{% Ln}z_{2}.$
This is interpreted that every value of $\operatorname{Ln}\left(z_{1}z_{2}\right)$ is one of the values of $\operatorname{Ln}z_{1}+\operatorname{Ln}z_{2}$, and vice versa. …
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: 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: 5.10 Continued Fractions
5.10.1 $\operatorname{Ln}\Gamma\left(z\right)+z-\left(z-\tfrac{1}{2}\right)\ln z-% \tfrac{1}{2}\ln\left(2\pi\right)=\cfrac{a_{0}}{z+\cfrac{a_{1}}{z+\cfrac{a_{2}}% {z+\cfrac{a_{3}}{z+\cfrac{a_{4}}{z+\cfrac{a_{5}}{z+}}}}}}\cdots,$
##### 6: 4.38 Inverse Hyperbolic Functions: Further Properties
4.38.15 $\operatorname{Arcsinh}u\pm\operatorname{Arcsinh}v=\operatorname{Arcsinh}\left(% u(1+v^{2})^{1/2}\pm v(1+u^{2})^{1/2}\right),$
4.38.16 $\operatorname{Arccosh}u\pm\operatorname{Arccosh}v=\operatorname{Arccosh}\left(% uv\pm((u^{2}-1)(v^{2}-1))^{1/2}\right),$
4.38.17 $\operatorname{Arctanh}u\pm\operatorname{Arctanh}v=\operatorname{Arctanh}\left(% \frac{u\pm v}{1\pm uv}\right),$
4.38.18 $\operatorname{Arcsinh}u\pm\operatorname{Arccosh}v=\operatorname{Arcsinh}\left(% uv\pm((1+u^{2})(v^{2}-1))^{1/2}\right)=\operatorname{Arccosh}\left(v(1+u^{2})^% {1/2}\pm u(v^{2}-1)^{1/2}\right),$
4.38.19 $\operatorname{Arctanh}u\pm\operatorname{Arccoth}v=\operatorname{Arctanh}\left(% \frac{uv\pm 1}{v\pm u}\right)=\operatorname{Arccoth}\left(\frac{v\pm u}{uv\pm 1% }\right).$
##### 7: 5.17 Barnes’ $G$-Function (Double Gamma Function)
5.17.4 $\operatorname{Ln}G\left(z+1\right)=\tfrac{1}{2}z\ln\left(2\pi\right)-\tfrac{1}% {2}z(z+1)+z\operatorname{Ln}\Gamma\left(z+1\right)-\int_{0}^{z}\operatorname{% Ln}\Gamma\left(t+1\right)\mathrm{d}t.$
In this equation (and in (5.17.5) below), the $\operatorname{Ln}$’s have their principal values on the positive real axis and are continued via continuity, as in §4.2(i). …
5.17.5 $\operatorname{Ln}G\left(z+1\right)\sim\tfrac{1}{4}z^{2}+z\operatorname{Ln}% \Gamma\left(z+1\right)-\left(\tfrac{1}{2}z(z+1)+\tfrac{1}{12}\right)% \operatorname{Ln}z-\ln A+\sum_{k=1}^{\infty}\frac{B_{2k+2}}{2k(2k+1)(2k+2)z^{2% k}}.$
##### 8: 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 cutIn all other cases, $z^{a}$ is a multivalued function with branch point at $z=0$. … This result is also valid when $z^{a}$ has its principal value, provided that the branch of $\operatorname{Ln}w$ satisfies …
##### 9: 4.7 Derivatives and Differential Equations
4.7.4 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\operatorname{Ln}z=(-1)^{n-1}(n-1)!z% ^{-n}.$
4.7.6 $w(z)=\operatorname{Ln}\left(f(z)\right)+\hbox{ constant}.$
When $a^{z}$ is a general power, $\ln a$ is replaced by the branch of $\operatorname{Ln}a$ used in constructing $a^{z}$. …
##### 10: 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 $z$. $\operatorname{Arctan}z$ and $\operatorname{Arccot}z$ have branch points at $z=\pm\mathrm{i}$; the other four functions have branch points at $z=\pm 1$. …