generalized exponential integral

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1: 8.19 Generalized Exponential Integral

§8.19 Generalized Exponential Integral

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Figure 8.19.5: $E_2\left(x+\mathrm{i}y\right)$, $-3\le x\le 3$, $-3\le y\le 3$. … Magnify 3D Help

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§8.19(v) Recurrence Relation and Derivatives

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§8.19(x) Integrals

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§8.19(xi) Further Generalizations

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2: 8.24 Physical Applications

§8.24 Physical Applications

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§8.24(iii) Generalized Exponential Integral

… ►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)).

3: 8.27 Approximations

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§8.27(ii) Generalized Exponential Integral

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Luke (1975, p. 103) gives Chebyshev-series expansions for $E_1\left(x\right)$ and related functions for $x\ge 5$.

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4: 8.26 Tables

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§8.26(iv) Generalized Exponential Integral

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

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§8.20(i) Large $z$

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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:

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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:

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See also:

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

… ►Where the sectors of validity of (8.20.2) and (8.20.3) overlap the contribution of the first term on the right-hand side of (8.20.3) is exponentially small compared to the other contribution; compare §2.11(ii). … ►

§8.20(ii) Large $p$

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6: 8.1 Special Notation

… ►Unless otherwise indicated, primes denote derivatives with respect to the argument. ►The functions treated in this chapter are the incomplete gamma functions $\gamma (a,z)$, $\mathrm{\Gamma }(a,z)$, ${\gamma }^{\ast }(a,z)$, $P(a,z)$, and $Q(a,z)$; the incomplete beta functions ${\mathrm{B}}_x(a,b)$ and $I_x(a,b)$; the generalized exponential integral $E_p\left(z\right)$; the generalized sine and cosine integrals $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$. …

7: 7.11 Relations to Other Functions

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Incomplete Gamma Functions and Generalized Exponential Integral

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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

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8: 8.22 Mathematical Applications

§8.22 Mathematical Applications

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

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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:

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See also:

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

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9: 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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10: 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:

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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:

pMML, png, TeX

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 . …