# fractional integrals

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##### 1: 1.15 Summability Methods
###### §1.15(vi) FractionalIntegrals
1.15.47 $I^{\alpha}f(x)=\frac{1}{\Gamma\left(\alpha\right)}\int^{x}_{0}(x-t)^{\alpha-1}% f(t)\,\mathrm{d}t.$
1.15.48 $I^{\alpha}I^{\beta}=I^{\alpha+\beta},$ $\Re\alpha>0$, $\Re\beta>0$.
###### §1.15(vii) Fractional Derivatives
1.15.52 $D^{k}I^{\alpha}=D^{n}I^{\alpha+n-k},$ $k=1,2,\dots,n$.
##### 3: 19.13 Integrals of Elliptic Integrals
Cvijović and Klinowski (1994) contains fractional integrals (with free parameters) for $F\left(\phi,k\right)$ and $E\left(\phi,k\right)$, together with special cases. …
##### 4: 2.6 Distributional Methods
###### §2.6(iii) FractionalIntegrals
The Riemann–Liouville fractional integral of order $\mu$ is defined by
2.6.33 $I^{\mu}f(x)=\frac{1}{\Gamma\left(\mu\right)}\int_{0}^{x}(x-t)^{\mu-1}f(t)\,% \mathrm{d}t,$ $\mu>0$;
2.6.48 $I^{\mu}f(x)=\frac{1}{\Gamma\left(\mu\right)}\int_{0}^{x}(x-t)^{\mu-1}t^{1-% \alpha}(1+t)^{-1}\,\mathrm{d}t,$
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). …
##### 6: 15.6 Integral Representations
Note that (15.6.8) can be rewritten as a fractional integral. …
##### 8: Bibliography K
• D. Karp, A. Savenkova, and S. M. Sitnik (2007) Series expansions for the third incomplete elliptic integral via partial fraction decompositions. J. Comput. Appl. Math. 207 (2), pp. 331–337.
• T. H. Koornwinder (2015) Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.
• ##### 9: Bibliography L
• E. R. Love (1972b) Two index laws for fractional integrals and derivatives. J. Austral. Math. Soc. 14, pp. 385–410.
• ##### 10: Bibliography M
• J. P. McClure and R. Wong (1979) Exact remainders for asymptotic expansions of fractional integrals. J. Inst. Math. Appl. 24 (2), pp. 139–147.