generalized exponentials and logarithms

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1: 8.24 Physical Applications

§8.24 Physical Applications

… ►The function $\gamma (a,x)$ appears in: discussions of power-law relaxation times in complex physical systems (Sornette (1998)); logarithmic oscillations in relaxation times for proteins (Metzler et al. (1999)); Gaussian orbitals and exponential (Slater) orbitals in quantum chemistry (Shavitt (1963), Shavitt and Karplus (1965)); population biology and ecological systems (Camacho et al. (2002)). … ►

§8.24(iii) Generalized Exponential Integral

►The function $E_p\left(x\right)$, with $p>0$, appears in theories of transport and radiative equilibrium (Hopf (1934), Kourganoff (1952), Altaç (1996)). ►With more general values of $p$, $E_p\left(x\right)$ supplies fundamental auxiliary functions that are used in the computation of molecular electronic integrals in quantum chemistry (Harris (2002), Shavitt (1963)), and also wave acoustics of overlapping sound beams (Ding (2000)).

2: 4.44 Other Applications

§4.44 Other Applications

… ►For applications of generalized exponentials and generalized logarithms to computer arithmetic see §3.1(iv). …

3: 4.12 Generalized Logarithms and Exponentials

§4.12 Generalized Logarithms and Exponentials

►A generalized exponential function $\varphi (x)$ satisfies the equations ►

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

… ►

4.12.5 $$\varphi (x)=\psi (x)=x,$$ $0\le x\le 1$.

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Symbols:

$x$: real variable, $\varphi (z)$: generalized exponential and $\psi (x)$: generalized logarithm

Permalink:

http://dlmf.nist.gov/4.12.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.12 and Ch.4

… ►

4.12.6 $$\varphi (x)=\ln \left(x+1\right),$$ ,

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Symbols:

$\ln z$: principal branch of logarithm function, $x$: real variable and $\varphi (z)$: generalized exponential

Permalink:

http://dlmf.nist.gov/4.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.12 and Ch.4

…

4: 8.19 Generalized Exponential Integral

… ►

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

… ►

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

… ►

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

… ►

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

… ►

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

…

5: 8.20 Asymptotic Expansions of $E_p\left(z\right)$

… ►

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

ⓘ

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

… ►

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

ⓘ

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

… ►

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

ⓘ

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

… ►

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

ⓘ

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

…

6: 7.11 Relations to Other Functions

… ►

Incomplete Gamma Functions and Generalized Exponential Integral

… ►

7.11.3 $$\mathrm{erfc}z=\frac{z}{\sqrt{\pi }}{E}_{\frac{1}{2}}\left(z^2\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $E_p\left(z\right)$: generalized exponential integral and $z$: complex variable

Referenced by:

§7.12(i)

Permalink:

http://dlmf.nist.gov/7.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §7.11, §7.11 and Ch.7

… ►

7.11.4 $$\mathrm{erf}z=\frac{2z}{\sqrt{\pi }}M(\frac{1}{2},\frac{3}{2},-z^2)=\frac{2z}{\sqrt{\pi }}{\mathrm{e}}^{-z^2}M(1,\frac{3}{2},z^2),$$

ⓘ

Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

A&S Ref:

7.1.21

Permalink:

http://dlmf.nist.gov/7.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §7.11, §7.11 and Ch.7

►

7.11.5 $$\mathrm{erfc}z=\frac{1}{\sqrt{\pi }}{\mathrm{e}}^{-z^2}U(\frac{1}{2},\frac{1}{2},z^2)=\frac{z}{\sqrt{\pi }}{\mathrm{e}}^{-z^2}U(1,\frac{3}{2},z^2).$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

Referenced by:

§7.11

Permalink:

http://dlmf.nist.gov/7.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §7.11, §7.11 and Ch.7

… ►

Generalized Hypergeometric Functions

…

7: 8.22 Mathematical Applications

§8.22 Mathematical Applications

… ►

8.22.1 $$F_p\left(z\right)=\frac{\mathrm{\Gamma }\left(p\right)}{2\pi }{z}^{1-p}{E}_p\left(z\right)=\frac{\mathrm{\Gamma }\left(p\right)}{2\pi }\mathrm{\Gamma }(1-p,z),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $F_p\left(z\right)$: terminant function, $z$: complex variable and $p$: parameter

Permalink:

http://dlmf.nist.gov/8.22.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.22(i), §8.22 and Ch.8

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8.22.2 $${\zeta }_x(s)=\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^x\frac{t^{s-1}}{{\mathrm{e}}^t-1}dt,$$ $\mathrm{\Re }s>1$,

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Defines:

${\zeta }_x(s)$: incomplete Riemann zeta function (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable

Permalink:

http://dlmf.nist.gov/8.22.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.22(ii), §8.22 and Ch.8

… ►The Debye functions ${\int }_0^x{t}^n{\left({\mathrm{e}}^t-1\right)}^{-1}dt$ and ${\int }_x^{\mathrm{\infty }}{t}^n{\left({\mathrm{e}}^t-1\right)}^{-1}dt$ are closely related to the incomplete Riemann zeta function and the Riemann zeta function. …

8: 2.11 Remainder Terms; Stokes Phenomenon

… ►

2.11.5 $$E_p\left(z\right)=\frac{{\mathrm{e}}^{-z}{z}^{p-1}}{\mathrm{\Gamma }\left(p\right)}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-zt}{t}^{p-1}}{1+t}dt$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral and $\int$: integral

Referenced by:

§2.11(iii)

Permalink:

http://dlmf.nist.gov/2.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(ii), §2.11 and Ch.2

… ►

2.11.6 $$E_p\left(z\right)\sim \frac{{\mathrm{e}}^{-z}}{z}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(p\right)}_s}{z^s}$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm and $E_p\left(z\right)$: generalized exponential integral

Referenced by:

§2.11(ii), §2.11(iii), §2.11(iii), §2.11(iv), §2.11(iv)

Permalink:

http://dlmf.nist.gov/2.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(ii), §2.11 and Ch.2

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2.11.7 $$E_p\left(z\right)\sim \frac{2\pi \mathrm{i}{\mathrm{e}}^{-p\pi \mathrm{i}}}{\mathrm{\Gamma }\left(p\right)}{z}^{p-1}+\frac{{\mathrm{e}}^{-z}}{z}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(p\right)}_s}{z^s},$$

ⓘ

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 and $\mathrm{i}$: imaginary unit

Referenced by:

§2.11(ii), §2.11(iv), §2.11(iv), §2.11(iv)

Permalink:

http://dlmf.nist.gov/2.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(ii), §2.11 and Ch.2

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2.11.10 $$E_p\left(z\right)=\frac{{\mathrm{e}}^{-z}}{z}\sum _{s=0}^{n-1}{(-1)}^s\frac{{\left(p\right)}_s}{z^s}+{(-1)}^n\frac{2\pi }{\mathrm{\Gamma }\left(p\right)}{z}^{p-1}{F}_{n+p}\left(z\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: 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 and $F_p\left(z\right)$: terminant function

Referenced by:

§2.11(iii), §2.11(iv)

Permalink:

http://dlmf.nist.gov/2.11.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(iii), §2.11 and Ch.2

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2.11.11 $$F_{n+p}\left(z\right)=\frac{{\mathrm{e}}^{-z}}{2\pi }{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-zt}{t}^{n+p-1}}{1+t}dt=\frac{\mathrm{\Gamma }\left(n+p\right)}{2\pi }\frac{E_{n+p}\left(z\right)}{z^{n+p-1}}.$$

ⓘ

Defines:

$F_p\left(z\right)$: terminant function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral and $\int$: integral

Permalink:

http://dlmf.nist.gov/2.11.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(iii), §2.11 and Ch.2

…

9: Bibliography C

… ►

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

…

10: 8.4 Special Values

… ►For $E_n\left(z\right)$ see §8.19(i). … ►

8.4.5 $$\mathrm{\Gamma }(1,z)={\mathrm{e}}^{-z},$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/8.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.4 and Ch.8

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8.4.9 $$P(n+1,z)=1-{\mathrm{e}}^{-z}{e}_n(z),$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $P(a,z)$: normalized incomplete gamma function, $z$: complex variable, $n$: nonnegative integer and $e_n(z)$: functions

A&S Ref:

6.5.13

Permalink:

http://dlmf.nist.gov/8.4.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §8.4 and Ch.8

►

8.4.10 $$Q(n+1,z)={\mathrm{e}}^{-z}{e}_n(z),$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $Q(a,z)$: normalized incomplete gamma function, $z$: complex variable, $n$: nonnegative integer and $e_n(z)$: functions

Permalink:

http://dlmf.nist.gov/8.4.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §8.4 and Ch.8

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8.4.13 $$\mathrm{\Gamma }(1-n,z)=z^{1-n}{E}_n\left(z\right),$$

ⓘ

Symbols:

$E_p\left(z\right)$: generalized exponential integral, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§8.19(iv)

Permalink:

http://dlmf.nist.gov/8.4.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §8.4 and Ch.8

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