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21: 8.19 Generalized Exponential Integral

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8.19.1 $$E_p\left(z\right)=z^{p-1}\mathrm{\Gamma }(1-p,z).$$

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Defines:

$E_p\left(z\right)$: generalized exponential integral

Symbols:

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

A&S Ref:

5.1.45 6.5.9 (Definition extended to general values of $p$.)

Referenced by:

§8.19(i), §8.19(iii), §8.19(iv), §8.19(v), §8.19(vi), §8.19(vii)

Permalink:

http://dlmf.nist.gov/8.19.E1

Encodings:

pMML, png, TeX

See also:

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

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8.19.6 $$E_p\left(0\right)=\frac{1}{p-1},$$ $\mathrm{\Re }p>1$,

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Symbols:

$E_p\left(z\right)$: generalized exponential integral, $\mathrm{\Re }$: real part and $p$: parameter

A&S Ref:

5.1.23 (Extended to general values of $p$.)

Permalink:

http://dlmf.nist.gov/8.19.E6

Encodings:

pMML, png, TeX

See also:

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

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8.19.13 $$\frac{d}{dz}{E}_p\left(z\right)=-E_{p-1}\left(z\right),$$

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Symbols:

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

A&S Ref:

5.1.36 (Extended to general values of $p$.)

Referenced by:

§8.19(x)

Permalink:

http://dlmf.nist.gov/8.19.E13

Encodings:

pMML, png, TeX

See also:

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

… ►The general function $E_p\left(z\right)$ is attained by extending the path in (8.19.2) across the negative real axis. …

22: 14.21 Definitions and Basic Properties

… ►Many of the properties stated in preceding sections extend immediately from the $x$-interval $(1,\mathrm{\infty })$ to the cut $z$-plane $ℂ\backslash (-\mathrm{\infty },1]$. …

23: 17.12 Bailey Pairs

… ►The Bailey pair and Bailey chain concepts have been extended considerably. …

24: 27.4 Euler Products and Dirichlet Series

… ►In this case the infinite product on the right (extended over all primes $p$) is also absolutely convergent and is called the Euler product of the series. …

25: 9.11 Products

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9.11.4 $${\mathrm{Ai}}^2\left(z\right)+{\mathrm{Bi}}^2\left(z\right)=\frac{1}{{\pi }^{3/2}}{\int }_0^{\mathrm{\infty }}\exp \left(zt-\frac{1}{12}{t}^3\right)t^{-1/2}dt.$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral and $z$: complex variable

Source:

Muldoon (1977, p. 32, extended to complex $z$ by analytic continuation)

Referenced by:

§9.5(ii)

Permalink:

http://dlmf.nist.gov/9.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §9.11(iii), §9.11 and Ch.9

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9.11.19 $${\int }_0^{\mathrm{\infty }}\frac{dt}{{\mathrm{Ai}}^2\left(t\right)+{\mathrm{Bi}}^2\left(t\right)}={\int }_0^{\mathrm{\infty }}\frac{tdt}{\mathrm{Ai}_{}^{\prime }{}_{}{}^2\left(t\right)+\mathrm{Bi}_{}^{\prime }{}_{}{}^2\left(t\right)}=\frac{{\pi }^2}{6}.$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $M\left(z\right)$: Airy modulus function, $N\left(z\right)$: Airy modulus function, $\varphi \left(z\right)$: Airy phase function, $\theta \left(z\right)$: Airy phase function and $x$: real variable

Proof sketch:

Extend the definitions of §9.8(i) to positive values of $x$, obtain the indefinite integrals of $1/M^2\left(x\right)$ and $x/N^2\left(x\right)$ via the first two of (9.8.14), then combine the values of $\theta \left(0\right)$ and $\varphi \left(0\right)$ given in §9.8(i) with $\theta \left(+\mathrm{\infty }\right)=\varphi \left(+\mathrm{\infty }\right)=0$ obtained from (9.8.4), (9.8.8), and §9.7(ii). (Communicated by M.E. Muldoon.)

Permalink:

http://dlmf.nist.gov/9.11.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §9.11(v), §9.11 and Ch.9

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26: 4.40 Integrals

… ►The results in §§4.40(ii) and 4.40(iv) can be extended to the complex plane by using continuous branches and avoiding singularities. …

27: 27.13 Functions

… ►Also, Milne (1996, 2002) announce new infinite families of explicit formulas extending Jacobi’s identities. …

28: Bibliography G

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On mean convergence of extended Lagrange interpolation. J. Comput. Appl. Math. 43 (1-2), pp. 19–35.

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Notes:

Orthogonal polynomials and numerical methods

External Links:

ISSN 0377-0427, Document, Link, MathReview (Mircea Ivan), ZentralBlatt

Referenced by:

Erratum (V1.0.28) for Table 18.3.1.

Encodings:

BibTeX

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An extended class of orthogonal polynomials defined by a Sturm-Liouville problem. J. Math. Anal. Appl. 359 (1), pp. 352–367.

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External Links:

ISSN 0022-247X, Document, Link, MathReview (Li Zhong Peng)

Referenced by:

§18.36(vi).

Encodings:

BibTeX

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29: 4.2 Definitions

… ►Most texts extend the definition of the principal value to include the branch cut … ►As a consequence, it has the advantage of extending regions of validity of properties of principal values. …

30: 10.41 Asymptotic Expansions for Large Order

… ►The expansions (10.41.3)–(10.41.6) also hold uniformly in the sector $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$ , with the branches of the fractional powers in (10.41.3)–(10.41.8) extended by continuity from the positive real $z$-axis. … ►We then extend the validity of this property from $z\to \pm \mathrm{i}\mathrm{\infty }$ to $z\to \mathrm{\infty }$ in the sector $-\pi +\delta \le \mathrm{ph}z\le 2\pi -\delta$ in the case of $H_{\nu }^{(1)}\left(\nu z\right)$, and to $z\to \mathrm{\infty }$ in the sector $-2\pi +\delta \le \mathrm{ph}z\le \pi -\delta$ in the case of $H_{\nu }^{(2)}\left(\nu z\right)$. …