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1: 1.15 Summability Methods
§1.15(vi) Fractional Integrals
1.15.47 I α f ( x ) = 1 Γ ( α ) 0 x ( x - t ) α - 1 f ( t ) d t .
1.15.48 I α I β = I α + β , α > 0 , β > 0 .
§1.15(vii) Fractional Derivatives
1.15.52 D k I α = D n I α + n - k , k = 1 , 2 , , n .
2: 6.9 Continued Fraction
§6.9 Continued Fraction
3: 19.13 Integrals of Elliptic Integrals
Cvijović and Klinowski (1994) contains fractional integrals (with free parameters) for F ( ϕ , k ) and E ( ϕ , k ) , together with special cases. …
4: 2.6 Distributional Methods
§2.6(iii) Fractional Integrals
The Riemann–Liouville fractional integral of order μ is defined by
2.6.33 I μ f ( x ) = 1 Γ ( μ ) 0 x ( x - t ) μ - 1 f ( t ) d t , μ > 0 ;
2.6.35 I μ f ( x ) = 1 Γ ( μ ) ( t μ - 1 f ) ( x ) .
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 g ; see Li and Wong (1994). …
5: 18.17 Integrals
§18.17(iv) Fractional Integrals
Jacobi
Laguerre
6: 15.6 Integral Representations
Note that (15.6.8) can be rewritten as a fractional integral. …
7: 10.43 Integrals
§10.43(iii) Fractional Integrals
§10.43(iv) Integrals over the Interval ( 0 , )
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.