# multivalued

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##### 12: 1.10 Functions of a Complex Variable
###### §1.10(vi) Multivalued Functions
Let $F(z)$ be a multivalued function and $D$ be a domain. … Branches can be constructed in two ways: …
##### 13: 22.14 Integrals
22.14.2 $\int\operatorname{cn}\left(x,k\right)\mathrm{d}x=k^{-1}\operatorname{Arccos}% \left(\operatorname{dn}\left(x,k\right)\right),$
22.14.3 $\int\operatorname{dn}\left(x,k\right)\mathrm{d}x=\operatorname{Arcsin}\left(% \operatorname{sn}\left(x,k\right)\right)=\operatorname{am}\left(x,k\right).$
22.14.5 $\int\operatorname{sd}\left(x,k\right)\mathrm{d}x=(kk^{\prime})^{-1}% \operatorname{Arcsin}\left(-k\operatorname{cd}\left(x,k\right)\right),$
22.14.6 $\int\operatorname{nd}\left(x,k\right)\mathrm{d}x={k^{\prime}}^{-1}% \operatorname{Arccos}\left(\operatorname{cd}\left(x,k\right)\right).$
##### 14: 5.9 Integral Representations
5.9.10 $\operatorname{Ln}\Gamma\left(z\right)=\left(z-\tfrac{1}{2}\right)\ln z-z+% \tfrac{1}{2}\ln\left(2\pi\right)+2\int_{0}^{\infty}\frac{\operatorname{arctan}% \left(t/z\right)}{e^{2\pi t}-1}\mathrm{d}t,$
5.9.11 $\operatorname{Ln}\Gamma\left(z+1\right)=-\gamma z-\frac{1}{2\pi i}\int_{-c-% \infty i}^{-c+\infty i}\frac{\pi z^{-s}}{s\sin\left(\pi s\right)}\zeta\left(-s% \right)\mathrm{d}s,$
##### 15: 5.11 Asymptotic Expansions
5.11.1 $\operatorname{Ln}\Gamma\left(z\right)\sim\left(z-\tfrac{1}{2}\right)\ln z-z+% \tfrac{1}{2}\ln\left(2\pi\right)+\sum_{k=1}^{\infty}\frac{B_{2k}}{2k(2k-1)z^{2% k-1}}$
5.11.8 $\operatorname{Ln}\Gamma\left(z+h\right)\sim\left(z+h-\tfrac{1}{2}\right)\ln z-% z+\tfrac{1}{2}\ln\left(2\pi\right)+\sum_{k=2}^{\infty}\frac{(-1)^{k}B_{k}\left% (h\right)}{k(k-1)z^{k-1}},$
##### 17: 14.25 Integral Representations
where the multivalued functions have their principal values when $1 and are continuous in $\mathbb{C}\setminus(-\infty,1]$. …
##### 18: 19.21 Connection Formulas
19.21.7 $(x-y)R_{D}\left(y,z,x\right)+(z-y)R_{D}\left(x,y,z\right)=3R_{F}\left(x,y,z% \right)-3y^{1/2}x^{-1/2}z^{-1/2},$
19.21.8 $R_{D}\left(y,z,x\right)+R_{D}\left(z,x,y\right)+R_{D}\left(x,y,z\right)=3x^{-1% /2}y^{-1/2}z^{-1/2},$
19.21.10 $2R_{G}\left(x,y,z\right)=zR_{F}\left(x,y,z\right)-\tfrac{1}{3}(x-z)(y-z)R_{D}% \left(x,y,z\right)+3x^{1/2}y^{1/2}z^{-1/2},$ $z\neq 0$.
##### 19: 14.1 Special Notation
Multivalued functions take their principal values (§4.2(i)) unless indicated otherwise. …
##### 20: 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). …