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## 1—10 of 393 matching pages

##### 1: 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 $\operatorname{Ln}\frac{z_{1}}{z_{2}}=\operatorname{Ln}z_{1}-\operatorname{Ln}z% _{2},$
where the integer $k$ is chosen so that $\Re\left(-\mathrm{i}z\ln a\right)+2k\pi\in[-\pi,\pi]$. …
##### 3: 4.12 Generalized Logarithms and Exponentials
4.12.1 $\phi(x+1)=e^{\phi(x)},$ $-1,
4.12.6 $\phi(x)=\ln\left(x+1\right),$ $-1,
4.12.8 $\psi(x)=e^{x}-1,$ $-\infty,
4.12.9 $\psi(x)=\ell+\underbrace{\ln\cdots\ln}_{\ell\text{ times}}x,$ $x>1$,
4.12.10 $0\leq\underbrace{\ln\cdots\ln}_{\ell\text{times}}x<1.$
##### 4: 5.10 Continued Fractions
5.10.1 $\operatorname{Ln}\Gamma\left(z\right)+z-\left(z-\tfrac{1}{2}\right)\ln z-% \tfrac{1}{2}\ln\left(2\pi\right)=\cfrac{a_{0}}{z+\cfrac{a_{1}}{z+\cfrac{a_{2}}% {z+\cfrac{a_{3}}{z+\cfrac{a_{4}}{z+\cfrac{a_{5}}{z+}}}}}}\cdots,$
##### 5: 4.5 Inequalities
4.5.1 $\frac{x}{1+x}<\ln\left(1+x\right) $x>-1$, $x\neq 0$,
4.5.2 $x<-\ln\left(1-x\right)<\frac{x}{1-x},$ $x<1$, $x\neq 0$,
4.5.4 $\ln x\leq x-1,$ $x>0$,
4.5.5 $\ln x\leq a(x^{1/a}-1),$ $a$, $x>0$,
4.5.6 $|\ln\left(1+z\right)|\leq-\ln\left(1-|z|\right),$ $|z|<1$.
##### 6: 4.2 Definitions
The general logarithm function $\operatorname{Ln}z$ is defined by … The real and imaginary parts of $\ln z$ are given by … The only zero of $\ln z$ is at $z=1$. … Consequently $\ln z$ is two-valued on the cut, and discontinuous across the cut. … Natural logarithms have as base the unique positive number …
##### 8: 6.15 Sums
6.15.1 $\sum_{n=1}^{\infty}\operatorname{Ci}\left(\pi n\right)=\tfrac{1}{2}(\ln 2-% \gamma),$
6.15.2 $\sum_{n=1}^{\infty}\frac{\operatorname{si}\left(\pi n\right)}{n}=\tfrac{1}{2}% \pi(\ln\pi-1),$
6.15.3 $\sum_{n=1}^{\infty}(-1)^{n}\operatorname{Ci}\left(2\pi n\right)=1-\ln 2-\tfrac% {1}{2}\gamma,$
6.15.4 $\sum_{n=1}^{\infty}(-1)^{n}\frac{\operatorname{si}\left(2\pi n\right)}{n}=\pi(% \tfrac{3}{2}\ln 2-1).$
##### 9: 4.3 Graphics
Figure 4.3.2 illustrates the conformal mapping of the strip $-\pi<\Im z<\pi$ onto the whole $w$-plane cut along the negative real axis, where $w=e^{z}$ and $z=\ln w$ (principal value). …
##### 10: 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 $\operatorname{Ln}G\left(z+1\right)\sim\tfrac{1}{4}z^{2}+z\operatorname{Ln}% \Gamma\left(z+1\right)-\left(\tfrac{1}{2}z(z+1)+\tfrac{1}{12}\right)\ln z-\ln A% +\sum_{k=1}^{\infty}\frac{B_{2k+2}}{2k(2k+1)(2k+2)z^{2k}}.$
5.17.6 $A=e^{C}=1.28242\;71291\;00622\;63687\;\ldots,$
5.17.7 $C=\lim_{n\to\infty}\left(\sum_{k=1}^{n}k\ln k-\left(\tfrac{1}{2}n^{2}+\tfrac{1% }{2}n+\tfrac{1}{12}\right)\ln n+\tfrac{1}{4}n^{2}\right)=\frac{\gamma+\ln\left% (2\pi\right)}{12}-\frac{\zeta'\left(2\right)}{2\pi^{2}}=\frac{1}{12}-\zeta'% \left(-1\right),$