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is the fractional derivative (if q > 0) or fractional integral (if q < 0). If q = 0, then the q-th differintegral of a function is the function itself. In the context of fractional integration and differentiation, there are several definitions of the differintegral.
In mathematics, an Erdélyi–Kober operator is a fractional integration operation introduced by Arthur Erdélyi and Hermann Kober . The Erdélyi–Kober fractional integral is given by x − ν − α + 1 Γ ( α ) ∫ 0 x ( t − x ) α − 1 t − α − ν f ( t ) d t {\displaystyle {\frac {x^{-\nu -\alpha +1}}{\Gamma (\alpha )}}\int _{0 ...
for < and >.. These are the fractional generalizations of the -fold left- and right-integrals of the form ()and for ,respectively. Even though the integral operators in question are close resemblance of the famous Erdélyi–Kober operator, it is not possible to obtain the Hadamard fractional integrals as a direct consequence of the Erdélyi–Kober operators.
The theory of fractional integration for periodic functions (therefore including the "boundary condition" of repeating after a period) is given by the Weyl integral. It is defined on Fourier series , and requires the constant Fourier coefficient to vanish (thus, it applies to functions on the unit circle whose integrals evaluate to zero).
The Cauchy formula for repeated integration, named after Augustin-Louis Cauchy, allows one to compress n antiderivatives of a function into a single integral (cf. Cauchy's formula). For non-integer n it yields the definition of fractional integrals and (with n < 0) fractional derivatives.
Sobolev's original proof of the Sobolev embedding theorem relied on the following, sometimes known as the Hardy–Littlewood–Sobolev fractional integration theorem. An equivalent statement is known as the Sobolev lemma in (Aubin 1982, Chapter 2). A proof is in (Stein 1970, Chapter V, §1.3). Let 0 < α < n and 1 < p < q < ∞.
Integration by parts is a heuristic rather than a purely mechanical process for solving integrals; given a single function to integrate, the typical strategy is to carefully separate this single function into a product of two functions u(x)v(x) such that the residual integral from the integration by parts formula is easier to evaluate than the ...
The following is a list of integrals (antiderivative functions) of rational functions. Any rational function can be integrated by partial fraction decomposition of the function into a sum of functions of the form: