# generalized logarithms

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##### 1: 4.44 Other Applications
###### §4.44 Other Applications
For applications of generalized exponentials and generalized logarithms to computer arithmetic see §3.1(iv). …
##### 2: 4.12 Generalized Logarithms and Exponentials
###### §4.12 GeneralizedLogarithms and Exponentials
A generalized exponential function $\phi(x)$ satisfies the equations …Its inverse $\psi(x)$ is called a generalized logarithm. … For $C^{\infty}$ generalized logarithms, see Walker (1991). For analytic generalized logarithms, see Kneser (1950).
##### 3: 5.10 Continued Fractions
5.10.1 $\operatorname{Ln}\Gamma\left(z\right)+z-\left(z-\tfrac{1}{2}\right)\ln z-% \tfrac{1}{2}\ln\left(2\pi\right)=\cfrac{a_{0}}{z+\cfrac{a_{1}}{z+\cfrac{a_{2}}% {z+\cfrac{a_{3}}{z+\cfrac{a_{4}}{z+\cfrac{a_{5}}{z+}}}}}}\cdots,$
##### 4: 4.2 Definitions
###### §4.2(i) The Logarithm
The general logarithm function $\operatorname{Ln}z$ is defined by
4.2.1 $\operatorname{Ln}z=\int_{1}^{z}\frac{\,\mathrm{d}t}{t},$ $z\neq 0$,
With this definition the general logarithm is given by …
##### 5: 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 $\operatorname{Ln}\left(\exp z\right)=z+2k\pi\mathrm{i},$ $k\in\mathbb{Z}$,
##### 6: 4.7 Derivatives and Differential Equations
4.7.4 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\operatorname{Ln}z=(-1)^{n-1}(n-1)!z% ^{-n}.$
For a nonvanishing analytic function $f(z)$, the general solution of the differential equation …
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 $a^{z}$. …
##### 7: 5.17 Barnes’ $G$-Function (Double Gamma Function)
5.17.4 $\operatorname{Ln}G\left(z+1\right)=\tfrac{1}{2}z\ln\left(2\pi\right)-\tfrac{1}% {2}z(z+1)+z\operatorname{Ln}\Gamma\left(z+1\right)-\int_{0}^{z}\operatorname{% Ln}\Gamma\left(t+1\right)\,\mathrm{d}t.$
5.17.5 $\operatorname{Ln}G\left(z+1\right)\sim\tfrac{1}{4}z^{2}+z\operatorname{Ln}% \Gamma\left(z+1\right)-\left(\tfrac{1}{2}z(z+1)+\tfrac{1}{12}\right)\ln z-\ln A% +\sum_{k=1}^{\infty}\frac{B_{2k+2}}{2k(2k+1)(2k+2)z^{2k}}.$
##### 8: 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 $\operatorname{Ln}\Gamma\left(z+1\right)=-\gamma z-\frac{1}{2\pi i}\int_{-c-% \infty i}^{-c+\infty i}\frac{\pi z^{-s}}{s\sin\left(\pi s\right)}\zeta\left(-s% \right)\,\mathrm{d}s,$
##### 9: 6.4 Analytic Continuation
6.4.1 $E_{1}\left(z\right)=\operatorname{Ein}\left(z\right)-\operatorname{Ln}z-\gamma;$
##### 10: 3.1 Arithmetics and Error Measures
To eliminate overflow or underflow in finite-precision arithmetic numbers are represented by using generalized logarithms $\ln_{\ell}(x)$ given by …