generalized exponentials

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11: 8.28 Software

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§8.28(vi) Generalized Exponential Integral for Real Argument and Integer Parameter

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§8.28(vii) Generalized Exponential Integral for Complex Argument and/or Parameter

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12: 17.17 Physical Applications

… ►It involves $q$-generalizations of exponentials and Laguerre polynomials, and has been applied to the problems of the harmonic oscillator and Coulomb potentials. …

13: 2.11 Remainder Terms; Stokes Phenomenon

… ►From §8.19(i) the generalized exponential integral is given by ►

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

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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

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

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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:

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See also:

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

… ►However, on combining (2.11.6) with the connection formula (8.19.18), with $m=1$, we derive … ►Two different asymptotic expansions in terms of elementary functions, (2.11.6) and (2.11.7), are available for the generalized exponential integral in the sector . …

14: Bibliography O

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Uniform, exponentially improved, asymptotic expansions for the generalized exponential integral. SIAM J. Math. Anal. 22 (5), pp. 1460–1474.

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External Links:

ISSN 0036-1410, Document, MathReview (Hans-Jürgen Glaeske), ZentralBlatt

Referenced by:

§2.11(iii), §2.11(iii), §8.11(i), §8.20(i).

Encodings:

BibTeX

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The Generalized Exponential Integral. In Approximation and Computation (West Lafayette, IN, 1993), R. V. M. Zahar (Ed.), International Series of Numerical Mathematics, Vol. 119, pp. 497–510.

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External Links:

MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§2.11(iii), §8.19(viii).

Encodings:

BibTeX

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15: Bibliography C

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A numerical method for generalized exponential integrals. Comput. Math. Appl. 14 (4), pp. 261–268.

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External Links:

ISSN 0898-1221, MathReview (R. P. Boas), ZentralBlatt

Referenced by:

§8.25(v).

Encodings:

BibTeX

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On the evaluation of generalized exponential integrals $E_v(x)$ . J. Comput. Phys. 78 (2), pp. 278–287.

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External Links:

ISSN 0021-9991, Document, MathReview, ZentralBlatt

Referenced by:

§8.25(v), 2nd item.

Encodings:

BibTeX

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On a Tricomi series representation for the generalized exponential integral. Internat. J. Comput. Math. 31, pp. 257–262.

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Notes:

Includes Fortran code

External Links:

Document, ZentralBlatt

Referenced by:

§8.28(vi).

Encodings:

BibTeX

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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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16: 8.4 Special Values

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

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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:

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See also:

Annotations for §8.4 and Ch.8

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17: 13.6 Relations to Other Functions

… ►When $a-b$ is an integer or $a$ is a positive integer the Kummer functions can be expressed as incomplete gamma functions (or generalized exponential integrals). … ►

13.6.6 $$U(a,a,z)=z^{1-a}U(1,2-a,z)=z^{1-a}{\mathrm{e}}^z{E}_a\left(z\right)={\mathrm{e}}^z\mathrm{\Gamma }(1-a,z).$$

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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, $\mathrm{\Gamma }(a,z)$: incomplete gamma function and $z$: complex variable

A&S Ref:

13.6.28 (in different form)

Permalink:

http://dlmf.nist.gov/13.6.E6

Encodings:

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See also:

Annotations for §13.6(ii), §13.6 and Ch.13

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18: 6.2 Definitions and Interrelations

… ►The generalized exponential integral $E_p\left(z\right)$, $p\in ℂ$, is treated in Chapter 8. …

19: 13.18 Relations to Other Functions

… ►When $\frac{1}{2}-\kappa \pm \mu$ is an integer the Whittaker functions can be expressed as incomplete gamma functions (or generalized exponential integrals). … ►

13.18.17 $$W_{\frac{1}{2}\alpha +\frac{1}{2}+n,\frac{1}{2}\alpha }\left(z\right)={(-1)}^n{\left(\alpha +1\right)}_n{M}_{\frac{1}{2}\alpha +\frac{1}{2}+n,\frac{1}{2}\alpha }\left(z\right)={(-1)}^{n}n!{\mathrm{e}}^{-\frac{1}{2}z}{z}^{\frac{1}{2}\alpha +\frac{1}{2}}{L}_n^{(\alpha )}\left(z\right).$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer and $z$: complex variable

Referenced by:

(33.14.14)

Permalink:

http://dlmf.nist.gov/13.18.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §13.18(v), §13.18(v), §13.18 and Ch.13

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20: 3.10 Continued Fractions

… ►For special functions see §5.10 (gamma function), §7.9 (error function), §8.9 (incomplete gamma functions), §8.17(v) (incomplete beta function), §8.19(vii) (generalized exponential integral), §§10.10 and 10.33 (quotients of Bessel functions), §13.6 (quotients of confluent hypergeometric functions), §13.19 (quotients of Whittaker functions), and §15.7 (quotients of hypergeometric functions). …