fractional integrals

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1: 1.15 Summability Methods

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§1.15(vi) Fractional Integrals

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1.15.47 $${𝐼}^{\alpha }f(x)=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_0^x{(x-t)}^{\alpha -1}f(t)dt.$$

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Defines:

${𝐼}^{\alpha }$: fractional integral (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ and $\int$: integral

Referenced by:

§1.15(vi), §1.15(vi), Erratum (V1.0.14) for Subsections 1.15(vi), 1.15(vii), 2.6(iii)

Permalink:

http://dlmf.nist.gov/1.15.E47

Encodings:

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See also:

Annotations for §1.15(vi), §1.15 and Ch.1

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1.15.48 $${𝐼}^{\alpha }{𝐼}^{\beta }={𝐼}^{\alpha +\beta },$$ $\mathrm{\Re }\alpha >0$, $\mathrm{\Re }\beta >0$.

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Symbols:

$\mathrm{\Re }$: real part and ${𝐼}^{\alpha }$: fractional integral

Referenced by:

§1.15(vi)

Permalink:

http://dlmf.nist.gov/1.15.E48

Encodings:

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See also:

Annotations for §1.15(vi), §1.15 and Ch.1

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§1.15(vii) Fractional Derivatives

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1.15.52 $${𝐷}^k{𝐼}^{\alpha }={𝐷}^n{𝐼}^{\alpha +n-k},$$ $k=1,2,\mathrm{\dots },n$.

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Symbols:

$k$: integer, $n$: nonnegative integer, ${𝐼}^{\alpha }$: fractional integral and ${𝐷}^{\alpha }$: fractional derivative

Referenced by:

§1.15(vii)

Permalink:

http://dlmf.nist.gov/1.15.E52

Encodings:

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See also:

Annotations for §1.15(vii), §1.15 and Ch.1

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2: 6.9 Continued Fraction

§6.9 Continued Fraction

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3: 19.13 Integrals of Elliptic Integrals

… ►Cvijović and Klinowski (1994) contains fractional integrals (with free parameters) for $F(\varphi ,k)$ and $E(\varphi ,k)$, together with special cases. …

4: 2.6 Distributional Methods

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§2.6(iii) Fractional Integrals

►The Riemann–Liouville fractional integral of order $\mu$ is defined by ►

2.6.33 $${𝐼}^{\mu }f(x)=\frac{1}{\mathrm{\Gamma }\left(\mu \right)}{\int }_0^x{(x-t)}^{\mu -1}f(t)dt,$$ $\mu >0$;

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Defines:

${𝐼}^{\mu }$: fractional integral (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $\mu$: order and $f(t)$: locally integrable function

Referenced by:

§2.6(iii)

Permalink:

http://dlmf.nist.gov/2.6.E33

Encodings:

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See also:

Annotations for §2.6(iii), §2.6 and Ch.2

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2.6.35 $${𝐼}^{\mu }f(x)=\frac{1}{\mathrm{\Gamma }\left(\mu \right)}(t^{\mu -1}\ast f)(x).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mu$: order, ${𝐼}^{\mu }$: fractional integral , $\ast$: convolution and $f(t)$: locally integrable function

Referenced by:

§2.6(iii)

Permalink:

http://dlmf.nist.gov/2.6.E35

Encodings:

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Annotations for §2.6(iii), §2.6 and Ch.2

… ►If both $f$ and $g$ in (2.6.34) have asymptotic expansions of the form (2.6.9), then the distribution method can also be used to derive an asymptotic expansion of the convolution $f\ast g$; see Li and Wong (1994). …

5: 18.17 Integrals

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§18.17(iv) Fractional Integrals

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Jacobi

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Laguerre

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6: 15.6 Integral Representations

… ►Note that (15.6.8) can be rewritten as a fractional integral. …

7: 10.43 Integrals

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§10.43(iii) Fractional Integrals

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§10.43(iv) Integrals over the Interval ($0,\mathrm{\infty }$)

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8: Bibliography K

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Series expansions for the third incomplete elliptic integral via partial fraction decompositions. J. Comput. Appl. Math. 207 (2), pp. 331–337.

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External Links:

ISSN 0377-0427, Document, MathReview (José Luis López), ZentralBlatt

Referenced by:

§19.12, §19.5.

Encodings:

BibTeX

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Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.

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External Links:

ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§15.14, §15.14.

Encodings:

BibTeX

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9: Bibliography L

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Two index laws for fractional integrals and derivatives. J. Austral. Math. Soc. 14, pp. 385–410.

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External Links:

Document, MathReview (C. Fox), ZentralBlatt

Referenced by:

§1.15(vi), §1.15(vii).

Encodings:

BibTeX

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10: Bibliography M

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Exact remainders for asymptotic expansions of fractional integrals. J. Inst. Math. Appl. 24 (2), pp. 139–147.

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External Links:

ISSN 0020-2932, Document, MathReview, ZentralBlatt

Referenced by:

§2.6(iii).

Encodings:

BibTeX

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