generalized logarithms

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11: 5.11 Asymptotic Expansions

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5.11.1 $$\mathrm{Ln}\mathrm{\Gamma }\left(z\right)\sim \left(z-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln \left(2\pi \right)+\sum _{k=1}^{\mathrm{\infty }}\frac{B_{2k}}{2k(2k-1)z^{2k-1}}$$

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Symbols:

$B_n$: Bernoulli numbers, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.1.40

Referenced by:

§5.11(i), §5.11(i), §5.11(ii), §5.11(ii), Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8)

Permalink:

http://dlmf.nist.gov/5.11.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.10):

To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\mathrm{Ln}\mathrm{\Gamma }\left(z\right)$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }\left(z\right)$ was used, where $\ln$ is the principal branch of the logarithm.

Suggested 2015-02-13 by Philippe Spindel

See also:

Annotations for §5.11(i), §5.11 and Ch.5

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5.11.8 $$\mathrm{Ln}\mathrm{\Gamma }\left(z+h\right)\sim \left(z+h-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln \left(2\pi \right)+\sum _{k=2}^{\mathrm{\infty }}\frac{{(-1)}^k{B}_k\left(h\right)}{k(k-1)z^{k-1}},$$

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $z$: complex variable

Referenced by:

§12.10(ii), (5.11.8), §5.11(i), §5.11(ii), Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8), Erratum (V1.0.11) for Equation (5.11.8)

Permalink:

http://dlmf.nist.gov/5.11.E8

Encodings:

pMML, png, TeX

Generalization (effective with 1.0.11):

Originally $h$ in (5.11.8) was unnecessarily restricted to lie in the interval $[0,1]$. In fact, $h$ may lie anywhere in the complex plane.

Suggested 2015-02-28 by Nico Temme

Addition (effective with 1.0.10):

To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\mathrm{Ln}\mathrm{\Gamma }\left(z+h\right)$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }\left(z+h\right)$ was used, where $\ln$ is the principal branch of the logarithm.

Suggested 2015-02-13 by Philippe Spindel

See also:

Annotations for §5.11(i), §5.11(i), §5.11 and Ch.5

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12: Bibliography W

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Infinitely differentiable generalized logarithmic and exponential functions. Math. Comp. 57 (196), pp. 723–733.

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External Links:

ISSN 0025-5718, FullText, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§4.12.

Encodings:

BibTeX

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13: 8.19 Generalized Exponential Integral

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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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8.19.15 $$\frac{{\partial }^j{E}_p\left(z\right)}{{\partial p}^j}={(-1)}^j{\int }_1^{\mathrm{\infty }}{(\ln t)}^j{t}^{-p}{\mathrm{e}}^{-zt}dt,$$ $\mathrm{\Re }z>0$.

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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, $\ln z$: principal branch of logarithm function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\mathrm{\Re }$: real part, $z$: complex variable, $p$: parameter and $j$: numbers

Permalink:

http://dlmf.nist.gov/8.19.E15

Encodings:

pMML, png, TeX

See also:

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

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8.19.16 $$E_p\left(z\right)=z^{p-1}{\mathrm{e}}^{-z}U(p,p,z).$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: 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.E16

Encodings:

pMML, png, TeX

See also:

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

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8.19.24 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}{E}_n\left(t\right)dt=\frac{{(-1)}^{n-1}}{a^n}\left(\ln \left(1+a\right)+\sum _{k=1}^{n-1}\frac{{(-1)}^k{a}^k}{k}\right),$$ $n=1,2,\mathrm{\dots }$, $\mathrm{\Re }a>-1$,

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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, $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $a$: parameter, $k$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

5.1.34

Referenced by:

§8.19(x)

Permalink:

http://dlmf.nist.gov/8.19.E24

Encodings:

pMML, png, TeX

See also:

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

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14: 24.16 Generalizations

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24.16.1 $${\left(\frac{t}{{\mathrm{e}}^t-1}\right)}^{\mathrm{\ell }}{\mathrm{e}}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{B}_n^{(\mathrm{\ell })}\left(x\right)\frac{t^n}{n!},$$ ,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $B_n^{(\mathrm{\ell })}\left(x\right)$: generalized Bernoulli polynomials, $\mathrm{\ell }$: integer, $n$: integer, $t$: real or complex and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §24.16(i), §24.16(i), §24.16 and Ch.24

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24.16.2 $${\left(\frac{2}{{\mathrm{e}}^t+1}\right)}^{\mathrm{\ell }}{\mathrm{e}}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{E}_n^{(\mathrm{\ell })}\left(x\right)\frac{t^n}{n!},$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $E_n^{(\mathrm{\ell })}\left(x\right)$: generalized Euler polynomials, $\mathrm{\ell }$: integer, $n$: integer, $t$: real or complex and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §24.16(i), §24.16(i), §24.16 and Ch.24

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24.16.4 $${\left(\frac{\ln \left(1+t\right)}{t}\right)}^{\mathrm{\ell }}=\mathrm{\ell }\sum _{n=0}^{\mathrm{\infty }}\frac{B_n^{(\mathrm{\ell }+n)}}{\mathrm{\ell }+n}\frac{t^n}{n!},$$ .

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Symbols:

$!$: factorial (as in $n!$), $B_n^{(\mathrm{\ell })}\left(x\right)$: generalized Bernoulli polynomials, $\ln z$: principal branch of logarithm function, $\mathrm{\ell }$: integer, $n$: integer and $t$: real or complex

Referenced by:

§24.16(i)

Permalink:

http://dlmf.nist.gov/24.16.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §24.16(i), §24.16(i), §24.16 and Ch.24

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24.16.9 $${\left(\frac{t}{{\mathrm{e}}^t-1}\right)}^x=\sum _{n=0}^{\mathrm{\infty }}{B}_n^{(x)}\frac{t^n}{n!},$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $B_n^{(\mathrm{\ell })}\left(x\right)$: generalized Bernoulli polynomials, $n$: integer, $t$: real or complex and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.16.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.16(i), §24.16(i), §24.16 and Ch.24

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24.16.10 $$\sum _{a=1}^f\frac{\chi (a)t{\mathrm{e}}^{at}}{{\mathrm{e}}^{ft}-1}=\sum _{n=0}^{\mathrm{\infty }}{B}_{n,\chi }\frac{t^n}{n!},$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: integer, $t$: real or complex, $\chi (a)$: Dirichlet character, $f$: conductor and $B_{n,\chi }$: Generalized Bernoulli numbers

Permalink:

http://dlmf.nist.gov/24.16.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §24.16(ii), §24.16 and Ch.24

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15: 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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16: 5.18 $q$-Gamma and $q$-Beta Functions

… ►For generalized asymptotic expansions of $\ln {\mathrm{\Gamma }}_q\left(z\right)$ as $|z|\to \mathrm{\infty }$ see Olde Daalhuis (1994) and Moak (1984). …

17: 14.15 Uniform Asymptotic Approximations

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14.15.4 $${𝖯}_{\nu }^{-\mu }\left(x\right)=\frac{1}{\mathrm{\Gamma }\left(\mu +1\right)}{\left(1-{\alpha }^2\right)}^{-\mu /2}{\left(\frac{1-\alpha }{1+\alpha }\right)}^{(\nu /2)+(1/4)}{\left(\frac{p}{x}\right)}^{1/2}{\mathrm{e}}^{-\mu \rho }\left(1+O\left(\frac{1}{\mu }\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $\mathrm{e}$: base of natural logarithm, $x$: real variable, $\mu$: general order, $\nu$: general degree, $\alpha$ , $p$ and $\rho$

Referenced by:

§14.15(ii)

Permalink:

http://dlmf.nist.gov/14.15.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(ii), §14.15 and Ch.14

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14.15.9 $${𝑸}_{\nu }^{\mu }\left(x\right)={\left(\frac{\pi }{2}\right)}^{1/2}{\left(\frac{\mathrm{e}}{\mu }\right)}^{\nu +(1/2)}{\left(\frac{1-\alpha }{1+\alpha }\right)}^{\mu /2}{\left(1-{\alpha }^2\right)}^{-(\nu /2)-(1/4)}{\left(\frac{{\alpha }^2+{\eta }^2}{{\alpha }^2\left(x^2-1\right)+1}\right)}^{1/4}{I}_{\nu +\frac{1}{2}}\left(\mu \eta \right)\left(1+O\left(\frac{1}{\mu }\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $x$: real variable, $\mu$: general order, $\nu$: general degree, $\alpha$ and $\eta$

Permalink:

http://dlmf.nist.gov/14.15.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(ii), §14.15 and Ch.14

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14.15.20 $$\beta ={\mathrm{e}}^{\mu }{\left(\frac{\nu -\mu +\frac{1}{2}}{\nu +\mu +\frac{1}{2}}\right)}^{(\nu /2)+(1/4)}{\left({\left(\nu +\frac{1}{2}\right)}^2-{\mu }^2\right)}^{-\mu /2},$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\mu$: general order, $\nu$: general degree and $\beta$

Permalink:

http://dlmf.nist.gov/14.15.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(iv), §14.15 and Ch.14

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14.15.30 $${𝖯}_{\nu }^{-\mu }\left(x\right)=\frac{1}{{\left(\nu +\frac{1}{2}\right)}^{1/4}{2}^{(\nu +\mu )/2}\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +\frac{3}{4}\right)}{\left(\frac{{\zeta }^2+{\alpha }^2}{x^2+a^2}\right)}^{1/4}U(\mu -\nu -\frac{1}{2},{\left(2\nu +1\right)}^{1/2}\zeta )\left(1+O\left({\nu }^{-1}\ln \nu \right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $\ln z$: principal branch of logarithm function, $U(a,z)$: parabolic cylinder function, $x$: real variable, $\mu$: general order, $\nu$: general degree, $a$ , $\zeta$ and $\alpha$

Referenced by:

§14.15(v)

Permalink:

http://dlmf.nist.gov/14.15.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(v), §14.15 and Ch.14

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18: Errata

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To increase the regions of validity the logarithms of the gamma function that appears on their left-hand sides have all been changed to $\mathrm{Ln}\mathrm{\Gamma }(\cdot )$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }(\cdot )$ was used, where $\ln$ is the principal branch of the logarithm. These changes were recommended by Philippe Spindel on 2015-02-06.

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19: Bibliography C

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Generalized exponential and logarithmic functions. Comput. Math. Appl. Part B 12 (5-6), pp. 1091–1101.

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External Links:

ISSN 0886-9561, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§4.12.

Encodings:

BibTeX

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20: 14.24 Analytic Continuation

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14.24.1 $$P_{\nu }^{-\mu }\left(z{\mathrm{e}}^{s\pi \mathrm{i}}\right)={\mathrm{e}}^{s\nu \pi \mathrm{i}}{P}_{\nu }^{-\mu }\left(z\right)+\frac{2\mathrm{i}\sin \left(\left(\nu +\frac{1}{2}\right)s\pi \right){\mathrm{e}}^{-s\pi \mathrm{i}/2}}{\cos \left(\nu \pi \right)\mathrm{\Gamma }\left(\mu -\nu \right)}{𝑸}_{\nu }^{\mu }\left(z\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $z$: complex variable, $\mu$: general order, $\nu$: general degree and $s$: arbitrary integer

Referenced by:

§14.24, §14.24

Permalink:

http://dlmf.nist.gov/14.24.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.24 and Ch.14

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14.24.2 $${𝑸}_{\nu }^{\mu }\left(z{\mathrm{e}}^{s\pi \mathrm{i}}\right)={(-1)}^s{\mathrm{e}}^{-s\nu \pi \mathrm{i}}{𝑸}_{\nu }^{\mu }\left(z\right),$$

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Symbols:

${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $\mu$: general order, $\nu$: general degree and $s$: arbitrary integer

Referenced by:

§14.23, §14.24

Permalink:

http://dlmf.nist.gov/14.24.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.24 and Ch.14

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14.24.3 $$P_{\nu ,s}^{-\mu }\left(z\right)={\mathrm{e}}^{s\mu \pi \mathrm{i}}{P}_{\nu }^{-\mu }\left(z\right),$$

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $\mu$: general order, $\nu$: general degree and $s$: arbitrary integer

Permalink:

http://dlmf.nist.gov/14.24.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.24 and Ch.14

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14.24.4 $${𝑸}_{\nu ,s}^{\mu }\left(z\right)={\mathrm{e}}^{-s\mu \pi \mathrm{i}}{𝑸}_{\nu }^{\mu }\left(z\right)-\frac{\pi \mathrm{i}\sin \left(s\mu \pi \right)}{\sin \left(\mu \pi \right)\mathrm{\Gamma }\left(\nu -\mu +1\right)}{P}_{\nu }^{-\mu }\left(z\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $z$: complex variable, $\mu$: general order, $\nu$: general degree and $s$: arbitrary integer

Referenced by:

§14.24

Permalink:

http://dlmf.nist.gov/14.24.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.24 and Ch.14

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