fractional integrals

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11: 8.19 Generalized Exponential Integral

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§8.19(vii) Continued Fraction

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12: Errata

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A number of changes were made with regard to fractional integrals and derivatives. In §1.15(vi) a reference to Miller and Ross (1993) was added, the fractional integral operator of order $\alpha$ was more precisely identified as the Riemann-Liouville fractional integral operator of order $\alpha$, and a paragraph was added below (1.15.50) to generalize (1.15.47). In §1.15(vii) the sentence defining the fractional derivative was clarified. In §2.6(iii) the identification of the Riemann-Liouville fractional integral operator was made consistent with §1.15(vi).

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In §15.6, it was noted that (15.6.8) can be rewritten as a fractional integral.

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13: 10.22 Integrals

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Fractional Integral

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14: 19.5 Maclaurin and Related Expansions

… ►Coefficients of terms up to ${\lambda }^{49}$ are given in Lee (1990), along with tables of fractional errors in $K\left(k\right)$ and $E\left(k\right)$, $0.1\le k^2\le 0.9999$, obtained by using 12 different truncations of (19.5.6) in (19.5.8) and (19.5.9). …

15: Bibliography H

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Applied and Computational Complex Analysis. Vol. 2: Special Functions—Integral Transforms—Asymptotics—Continued Fractions. Wiley-Interscience [John Wiley & Sons], New York.

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Notes:

Reprinted in 1991.

External Links:

ISBN 0-471-01525-3, MathReview (A. E. Heins), ZentralBlatt

Referenced by:

§1.11(v), Ch.3.

Encodings:

BibTeX

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16: 25.11 Hurwitz Zeta Function

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25.11.27 $$\zeta (s,a)=\frac{1}{2}{a}^{-s}+\frac{a^{1-s}}{s-1}+\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\left(\frac{1}{{\mathrm{e}}^x-1}-\frac{1}{x}+\frac{1}{2}\right)x^{s-1}{\mathrm{e}}^{-ax}dx,$$ $\mathrm{\Re }s>-1$, $s\ne 1$, $\mathrm{\Re }a>0$.

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Symbols:

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

Keywords:

improper integral, integral representation

Proof sketch:

Derivable from (25.11.25) by first rewriting $(1-{\mathrm{e}}^{-x})={\mathrm{e}}^{-x}({\mathrm{e}}^x-1)$, then replacing $x^{s-1}\mapsto x^{s-1}(1+{\mathrm{e}}^{-x}-{\mathrm{e}}^{-x})$, using the identity ${\mathrm{e}}^{-x}=(1-{\mathrm{e}}^{-x})/({\mathrm{e}}^x-1)$, and replacing ${({\mathrm{e}}^x-1)}^{-1}\mapsto ({({\mathrm{e}}^x-1)}^{-1}-1/x+1/2-1/2+1/x)$ in the resulting integrand with (5.2.1), (5.5.1).

Referenced by:

(25.11.28)

Permalink:

http://dlmf.nist.gov/25.11.E27

Encodings:

pMML, png, TeX

Update (effective with 1.1.9):

The fraction in the integrand $\frac{x^{s-1}}{{\mathrm{e}}^{ax}}$ was rewritten to read $x^{s-1}{\mathrm{e}}^{-ax}$.

See also:

Annotations for §25.11(vii), §25.11 and Ch.25

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17: 7.18 Repeated Integrals of the Complementary Error Function

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§7.18(v) Continued Fraction

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18: Bibliography B

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Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind. IMA J. Numer. Anal. 12 (4), pp. 519–526.

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External Links:

ISSN 0272-4979, MathReview, ZentralBlatt

Referenced by:

§5.21, §5.24(ii).

Encodings:

BibTeX

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19: 7.22 Methods of Computation

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§7.22(i) Main Functions

►The methods available for computing the main functions in this chapter are analogous to those described in §§6.18(i)–6.18(iv) for the exponential integral and sine and cosine integrals, and similar comments apply. Additional references are Matta and Reichel (1971) for the application of the trapezoidal rule, for example, to the first of (7.7.2), and Gautschi (1970) and Cuyt et al. (2008) for continued fractions. ►

§7.22(ii) Goodwin–Staton Integral

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§7.22(iii) Repeated Integrals of the Complementary Error Function

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20: 19.29 Reduction of General Elliptic Integrals

… ►Partial fractions provide a reduction to integrals in which $𝐦$ has at most one nonzero component, and these are then reduced to basic integrals by the recurrence relations. …