general bases

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1: 4.2 Definitions

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§4.2(ii) Logarithms to a General Base $a$

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4.2.10 $${\log }_{a}b=\frac{1}{{\log }_{b}a}.$$

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Symbols:

${\log }_{a}z$: logarithm to general base and $a$: real or complex constant

A&S Ref:

4.1.20

Permalink:

http://dlmf.nist.gov/4.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §4.2(ii), §4.2 and Ch.4

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4.2.14 $${\log }_{\mathrm{e}}z=\ln z,$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, ${\log }_{a}z$: logarithm to general base, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.21

Permalink:

http://dlmf.nist.gov/4.2.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §4.2(ii), §4.2 and Ch.4

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4.2.16 $$\ln z=(\ln 10){\log }_{10}z,$$

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Symbols:

${\log }_{a}z$: logarithm to general base, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.23

Permalink:

http://dlmf.nist.gov/4.2.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §4.2(ii), §4.2 and Ch.4

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Powers with General Bases

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2: 4.12 Generalized Logarithms and Exponentials

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4.12.1 $$\varphi (x+1)={\mathrm{e}}^{\varphi (x)},$$ ,

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Symbols:

$\mathrm{e}$: base of natural logarithm, $x$: real variable and $\varphi (z)$: generalized exponential

Permalink:

http://dlmf.nist.gov/4.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.12 and Ch.4

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4.12.3 $$\psi ({\mathrm{e}}^x)=1+\psi (x),$$ ,

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Symbols:

$\mathrm{e}$: base of natural logarithm, $x$: real variable and $\psi (x)$: generalized logarithm

Permalink:

http://dlmf.nist.gov/4.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.12 and Ch.4

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4.12.8 $$\psi (x)={\mathrm{e}}^x-1,$$ ,

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Symbols:

$\mathrm{e}$: base of natural logarithm, $x$: real variable and $\psi (x)$: generalized logarithm

Permalink:

http://dlmf.nist.gov/4.12.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.12 and Ch.4

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3: 8.7 Series Expansions

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8.7.6 $$\mathrm{\Gamma }(a,x)=x^a{\mathrm{e}}^{-x}\sum _{n=0}^{\mathrm{\infty }}\frac{L_n^{(a)}\left(x\right)}{n+1},$$ $x>0$, .

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\mathrm{\Re }$: real part, $x$: real variable, $z$: complex variable, $a$: parameter and $n$: nonnegative integer

Proof sketch:

Integrate (18.12.13) from $z=0$ to $z=1$ and for the integral on the left-hand side use the substitution $z=\frac{t}{1+t}$. The resulting integral is (8.6.5). The constraint follows from (18.15.14).

Referenced by:

Erratum (V1.1.8) for Equation (8.7.6)

Permalink:

http://dlmf.nist.gov/8.7.E6

Encodings:

pMML, png, TeX

Addition (effective with 1.1.8):

The constraint was updated to include .

Suggested 2022-10-14 by Walter Gautschi

See also:

Annotations for §8.7 and Ch.8

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4: 3.1 Arithmetics and Error Measures

… ►To eliminate overflow or underflow in finite-precision arithmetic numbers are represented by using generalized logarithms ${\ln }_{\mathrm{\ell }}(x)$ given by …

5: 22 Jacobian Elliptic Functions

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6: 23 Weierstrass Elliptic and Modular
Functions

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7: 18.39 Applications in the Physical Sciences

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18.39.29 $${\psi }_{p,l}(r)=\sqrt{\frac{Zp!}{(p+2l+1)!}}\frac{{\mathrm{e}}^{-{\rho }_p/2}{\rho }_p^{l+1}}{p+l+1}{L}_p^{(2l+1)}\left({\rho }_p\right),$$ $p=0,1,2,\mathrm{\dots }$; $l=0,1,2,\mathrm{\dots }$,

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$) and $l$: nonnegative integer

Referenced by:

§18.39(ii), §18.39(ii), §18.39(ii)

Permalink:

http://dlmf.nist.gov/18.39.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

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18.39.34 $${\psi }_{n,l}(r)=\frac{1}{n}\sqrt{\frac{Z(n-l-1)!}{(n+l)!}}{\mathrm{e}}^{-{\rho }_n/2}{\rho }_n^{l+1}{L}_{n-l-1}^{(2l+1)}\left({\rho }_n\right),$$ $n=1,2,\mathrm{\dots },l=0,1,\mathrm{\dots }n-1$,

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $l$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.39.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

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18.39.37 $$R_{n,l}(r)=\frac{2}{n^2}\sqrt{\frac{Z^3(n-l-1)!}{(n+l)!}}{\mathrm{e}}^{-{\rho }_n/2}{\rho }_n^l{L}_{n-l-1}^{(2l+1)}\left({\rho }_n\right),$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $l$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§18.39(ii)

Permalink:

http://dlmf.nist.gov/18.39.E37

Encodings:

pMML, png, TeX

See also:

Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

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18.39.44 $${\varphi }_{n,l}(sr)={(sr)}^{l+1}{\mathrm{e}}^{-sr/2}{L}_n^{(2l+1)}\left(sr\right),$$ $n=0,1,2,\mathrm{\dots }$,

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$: base of natural logarithm, $l$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§18.39(iv), §18.39(iv), §18.39(iv)

Permalink:

http://dlmf.nist.gov/18.39.E44

Encodings:

pMML, png, TeX

See also:

Annotations for §18.39(iv), §18.39 and Ch.18

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8: 8.19 Generalized Exponential Integral

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8.19.2 $$E_p\left(z\right)=z^{p-1}{\int }_z^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t}}{t^p}dt.$$

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\int$: integral, $z$: complex variable and $p$: parameter

Keywords:

Laplace transform, Mellin transform

Referenced by:

§8.19(i), §8.19(viii)

Permalink:

http://dlmf.nist.gov/8.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(i), §8.19 and Ch.8

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8.19.5 $$E_0\left(z\right)=z^{-1}{\mathrm{e}}^{-z},$$ $z\ne 0$,

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Symbols:

$\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral and $z$: complex variable

A&S Ref:

5.1.24

Permalink:

http://dlmf.nist.gov/8.19.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(iii), §8.19 and Ch.8

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8.19.12 $$p{E}_{p+1}\left(z\right)+z{E}_p\left(z\right)={\mathrm{e}}^{-z}.$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $z$: complex variable and $p$: parameter

Permalink:

http://dlmf.nist.gov/8.19.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(v), §8.19 and Ch.8

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8.19.14 $$\frac{d}{dz}({\mathrm{e}}^z{E}_p\left(z\right))={\mathrm{e}}^z{E}_p\left(z\right)\left(1+\frac{p-1}{z}\right)-\frac{1}{z}.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $z$: complex variable and $p$: parameter

Permalink:

http://dlmf.nist.gov/8.19.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(v), §8.19 and Ch.8

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8.19.21

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Symbols:

$\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $x$: real variable and $n$: nonnegative integer

A&S Ref:

5.1.19

Permalink:

http://dlmf.nist.gov/8.19.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(ix), §8.19 and Ch.8

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9: 8.20 Asymptotic Expansions of $E_p\left(z\right)$

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8.20.1 $$E_p\left(z\right)=\frac{{\mathrm{e}}^{-z}}{z}\left(\sum _{k=0}^{n-1}{(-1)}^k\frac{{\left(p\right)}_k}{z^k}+{(-1)}^n\frac{{\left(p\right)}_n{\mathrm{e}}^z}{z^{n-1}}{E}_{n+p}\left(z\right)\right),$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $z$: complex variable, $p$: parameter, $k$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

5.1.51

Permalink:

http://dlmf.nist.gov/8.20.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.20(i), §8.20 and Ch.8

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8.20.2 $$E_p\left(z\right)\sim \frac{{\mathrm{e}}^{-z}}{z}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{{\left(p\right)}_k}{z^k},$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$,

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{ph}$: phase, $z$: complex variable, $p$: parameter, $k$: nonnegative integer and $\delta$: small positive constant

Referenced by:

§8.20(i)

Permalink:

http://dlmf.nist.gov/8.20.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.20(i), §8.20 and Ch.8

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8.20.3 $$E_p\left(z\right)\sim \pm \frac{2\pi \mathrm{i}}{\mathrm{\Gamma }\left(p\right)}{\mathrm{e}}^{\mp p\pi \mathrm{i}}{z}^{p-1}+\frac{{\mathrm{e}}^{-z}}{z}\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{\left(p\right)}_k}{z^k},$$ $\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}z\le \frac{7}{2}\pi -\delta$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $z$: complex variable, $p$: parameter, $k$: nonnegative integer and $\delta$: small positive constant

Referenced by:

§2.11(ii), §8.20(i)

Permalink:

http://dlmf.nist.gov/8.20.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.20(i), §8.20 and Ch.8

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8.20.6 $$E_p\left(\lambda p\right)\sim \frac{{\mathrm{e}}^{-\lambda p}}{(\lambda +1)p}\sum _{k=0}^{\mathrm{\infty }}\frac{A_k(\lambda )}{{(\lambda +1)}^{2k}}\frac{1}{p^k},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $p$: parameter, $k$: nonnegative integer and $A_k(\lambda )$

Permalink:

http://dlmf.nist.gov/8.20.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §8.20(ii), §8.20 and Ch.8

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10: 33.14 Definitions and Basic Properties

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33.14.14 $${\varphi }_{n,\mathrm{\ell }}(r)={(-1)}^{\mathrm{\ell }+1+n}{(2/n^3)}^{1/2}s(-1/n^2,\mathrm{\ell };r)=\frac{{(-1)}^{\mathrm{\ell }+1+n}}{n^{\mathrm{\ell }+2}}{\left(\frac{(n-\mathrm{\ell }-1)!}{(n+\mathrm{\ell })!}\right)}^{1/2}{(2r)}^{\mathrm{\ell }+1}{\mathrm{e}}^{-r/n}{L}_{n-\mathrm{\ell }-1}^{(2\mathrm{\ell }+1)}\left(2r/n\right)$$

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Defines:

${\varphi }_{n,\mathrm{\ell }}(r)$: function (locally)

Symbols:

$s(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer and $r$: real variable

Referenced by:

(33.14.14), (33.14.15), §33.14(iv), §33.22(v), Erratum (V1.0.24) for Equation (33.14.15), Erratum (V1.0.24) for Subsection 33.14(iv)

Permalink:

http://dlmf.nist.gov/33.14.E14

Encodings:

pMML, png, TeX

Addition (effective with 1.0.24):

For the second equation in (33.14.14) first, via (33.14.9) and (33.14.4), express ${\varphi }_{n,\mathrm{\ell }}(r)$ in terms of Whittaker functions and then use (13.18.17).

See also:

Annotations for §33.14(iv), §33.14 and Ch.33

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