# principal branch

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##### 1: 4.13 Lambert $W$-Function
We call the solution for which $W\left(x\right)\geq W\left(-1/e\right)$ the principal branch and denote it by $\mathrm{Wp}\left(x\right)$. …
4.13.5 $\mathrm{Wp}\left(x\right)=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{n^{n-2}}{(n-1)!}x% ^{n},$ $|x|<\dfrac{1}{e}$.
4.13.11 $\mathrm{Wm}\left(x\right)=-\eta-\ln\eta-\frac{\ln\eta}{\eta}+\frac{(\ln\eta)^{% 2}}{2\eta^{2}}-\frac{\ln\eta}{\eta^{2}}+O\left(\frac{(\ln\eta)^{3}}{\eta^{3}}% \right),$
##### 2: 6.15 Sums
6.15.1 $\sum_{n=1}^{\infty}\mathrm{Ci}\left(\pi n\right)=\tfrac{1}{2}(\ln 2-\gamma),$
6.15.2 $\sum_{n=1}^{\infty}\frac{\mathrm{si}\left(\pi n\right)}{n}=\tfrac{1}{2}\pi(\ln% \pi-1),$
6.15.3 $\sum_{n=1}^{\infty}(-1)^{n}\mathrm{Ci}\left(2\pi n\right)=1-\ln 2-\tfrac{1}{2}\gamma,$
6.15.4 $\sum_{n=1}^{\infty}(-1)^{n}\frac{\mathrm{si}\left(2\pi n\right)}{n}=\pi(\tfrac% {3}{2}\ln 2-1).$
##### 4: 4.1 Special Notation
 $k,m,n$ integers. …
##### 5: 4.6 Power Series
4.6.1 $\ln\left(1+z\right)=z-\tfrac{1}{2}z^{2}+\tfrac{1}{3}z^{3}-\cdots,$ $|z|\leq 1$, $z\neq-1$,
4.6.2 $\ln z=\left(\frac{z-1}{z}\right)+\frac{1}{2}\left(\frac{z-1}{z}\right)^{2}+% \frac{1}{3}\left(\frac{z-1}{z}\right)^{3}+\cdots,$ $\Re z\geq\frac{1}{2}$,
4.6.3 $\ln z=(z-1)-\tfrac{1}{2}(z-1)^{2}+\tfrac{1}{3}(z-1)^{3}-\cdots,$ $|z-1|\leq 1$, $z\neq 0$,
4.6.4 $\ln z=2\left(\left(\frac{z-1}{z+1}\right)+\frac{1}{3}\left(\frac{z-1}{z+1}% \right)^{3}+\frac{1}{5}\left(\frac{z-1}{z+1}\right)^{5}+\cdots\right),$ $\Re z\geq 0$, $z\neq 0$,
4.6.5 $\ln\left(\frac{z+1}{z-1}\right)=2\left(\frac{1}{z}+\frac{1}{3z^{3}}+\frac{1}{5% z^{5}}+\cdots\right),$ $|z|\geq 1$, $z\neq\pm 1$,
##### 6: 6.6 Power Series
6.6.3 $E_{1}\left(z\right)=-\ln z+e^{-z}\sum_{n=0}^{\infty}\frac{z^{n}}{n!}\psi\left(% n+1\right),$
##### 7: 4.5 Inequalities
4.5.1 $\frac{x}{1+x}<\ln\left(1+x\right) $x>-1$, $x\neq 0$,
4.5.2 $x<-\ln\left(1-x\right)<\frac{x}{1-x},$ $x<1$, $x\neq 0$,
4.5.3 $|\ln\left(1-x\right)|<\tfrac{3}{2}x,$ $0,
4.5.4 $\ln x\leq x-1,$ $x>0$,
4.5.5 $\ln x\leq a(x^{1/a}-1),$ $a$, $x>0$,
##### 8: 4.2 Definitions
This is a multivalued function of $z$ with branch point at $z=0$. The principal value, or principal branch, is defined by
4.2.2 $\ln z=\int_{1}^{z}\frac{\mathrm{d}t}{t},$
##### 9: 22.14 Integrals
22.14.1 $\int\operatorname{sn}\left(x,k\right)\mathrm{d}x=k^{-1}\ln\left(\operatorname{% dn}\left(x,k\right)-k\operatorname{cn}\left(x,k\right)\right),$
22.14.4 $\int\operatorname{cd}\left(x,k\right)\mathrm{d}x=k^{-1}\ln\left(\operatorname{% nd}\left(x,k\right)+k\operatorname{sd}\left(x,k\right)\right),$
22.14.7 $\int\operatorname{dc}\left(x,k\right)\mathrm{d}x=\ln\left(\operatorname{nc}% \left(x,k\right)+\operatorname{sc}\left(x,k\right)\right),$
22.14.8 $\int\operatorname{nc}\left(x,k\right)\mathrm{d}x={k^{\prime}}^{-1}\ln\left(% \operatorname{dc}\left(x,k\right)+k^{\prime}\operatorname{sc}\left(x,k\right)% \right),$
22.14.9 $\int\operatorname{sc}\left(x,k\right)\mathrm{d}x={k^{\prime}}^{-1}\ln\left(% \operatorname{dc}\left(x,k\right)+k^{\prime}\operatorname{nc}\left(x,k\right)% \right).$
##### 10: 4.8 Identities
4.8.12 $\ln\left(a^{z}\right)=z\ln a+2k\pi\mathrm{i},$
4.8.13 $\ln\left(a^{x}\right)=x\ln a,$ $a>0$.