# general base

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##### 1: 4.2 Definitions
###### §4.2(ii) Logarithms to a GeneralBase$a$
4.2.9 $\operatorname{log}_{a}z=\frac{\operatorname{log}_{b}z}{\operatorname{log}_{b}a},$
4.2.10 $\operatorname{log}_{a}b=\frac{1}{\operatorname{log}_{b}a}.$
##### 2: 4.12 Generalized Logarithms and Exponentials
4.12.1 $\phi(x+1)=e^{\phi(x)},$ $-1,
4.12.3 $\psi(e^{x})=1+\psi(x),$ $-\infty,
4.12.8 $\psi(x)=e^{x}-1,$ $-\infty,
##### 3: 8.7 Series Expansions
8.7.6 $\Gamma\left(a,x\right)=x^{a}e^{-x}\sum_{n=0}^{\infty}\frac{L^{(a)}_{n}\left(x% \right)}{n+1},$ $x>0$, $\Re a<\frac{1}{2}$.
##### 4: 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 …
##### 7: 18.39 Applications in the Physical Sciences
18.39.29 $\psi_{p,l}(r)=\sqrt{\frac{Zp!}{(p+2l+1)!}}\frac{{\mathrm{e}}^{-\rho_{p}/2}\rho% _{p}^{l+1}}{p+l+1}L^{(2l+1)}_{p}\left(\rho_{p}\right),$ $p=0,1,2,\dots$; $l=0,1,2,\dots$,
18.39.34 $\psi_{n,l}(r)=\frac{1}{n}\sqrt{\frac{Z(n-l-1)!}{(n+l)!}}{\mathrm{e}}^{-\rho_{n% }/2}\rho_{n}^{l+1}L^{(2l+1)}_{n-l-1}\left(\rho_{n}\right)\vphantom{argument},$ $n=1,2,\dots,l=0,1,\dots n-1$,
18.39.37 $R_{n,l}(r)=\frac{2}{n^{2}}\sqrt{\frac{Z^{3}(n-l-1)!}{(n+l)!}}{\mathrm{e}}^{-% \rho_{n}/2}\rho_{n}^{l}L^{(2l+1)}_{n-l-1}\left(\rho_{n}\right),$
##### 8: 8.19 Generalized Exponential Integral
8.19.2 $E_{p}\left(z\right)=z^{p-1}\int_{z}^{\infty}\frac{e^{-t}}{t^{p}}\,\mathrm{d}t.$
8.19.5 $E_{0}\left(z\right)=z^{-1}e^{-z},$ $z\neq 0$,
8.19.16 $E_{p}\left(z\right)=z^{p-1}e^{-z}U\left(p,p,z\right).$
##### 9: 8.20 Asymptotic Expansions of $E_{p}\left(z\right)$
8.20.1 $E_{p}\left(z\right)=\frac{e^{-z}}{z}\left(\sum_{k=0}^{n-1}(-1)^{k}\frac{{\left% (p\right)_{k}}}{z^{k}}+(-1)^{n}\frac{{\left(p\right)_{n}}e^{z}}{z^{n-1}}E_{n+p% }\left(z\right)\right),$ $n=1,2,3,\dots$.
8.20.2 $E_{p}\left(z\right)\sim\frac{e^{-z}}{z}\sum_{k=0}^{\infty}(-1)^{k}\frac{{\left% (p\right)_{k}}}{z^{k}},$ $|\operatorname{ph}z|\leq\frac{3}{2}\pi-\delta$,
8.20.3 $E_{p}\left(z\right)\sim\pm\frac{2\pi i}{\Gamma\left(p\right)}e^{\mp p\pi i}z^{% p-1}+\frac{e^{-z}}{z}\sum_{k=0}^{\infty}\frac{(-1)^{k}{\left(p\right)_{k}}}{z^% {k}},$ $\tfrac{1}{2}\pi+\delta\leq\pm\operatorname{ph}z\leq\tfrac{7}{2}\pi-\delta$,
8.20.6 $E_{p}\left(\lambda p\right)\sim\frac{e^{-\lambda p}}{(\lambda+1)p}\sum_{k=0}^{% \infty}\frac{A_{k}(\lambda)}{(\lambda+1)^{2k}}\frac{1}{p^{k}},$
##### 10: 33.14 Definitions and Basic Properties
33.14.14 $\phi_{n,\ell}(r)=(-1)^{\ell+1+n}(2/n^{3})^{1/2}s\left(-1/n^{2},\ell;r\right)=% \frac{(-1)^{\ell+1+n}}{n^{\ell+2}}\left(\frac{(n-\ell-1)!}{(n+\ell)!}\right)^{% 1/2}(2r)^{\ell+1}{\mathrm{e}}^{-r/n}L^{(2\ell+1)}_{n-\ell-1}\left(2r/n\right)$